SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES

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1 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES MATH 57A. Itroductio Our geometric ituitio is derived from three-dimesioal space. Three coordiates suffice. May objects of iterest i aalysis, however, require far more coordiates for a complete descriptio. For example, a fuctio f with domai [, ] is defied by ifiitely may coordiates f(t), oe for each t [, ]. Or, we could cosider f as beig determied by its Taylor series = a t (whe such a represetatio exists). I that case, the umbers a, a, a,... could be thought of as coordiates. Perhaps the associatio of Fourier coefficiets (there are coutably may of them) to a periodic fuctio is familiar; those are agai coordiates of a sort. Strage Thigs ca happe i ifiite dimesios. Oe usually meets these, gradually (reluctatly?), i a course o Real Aalysis or Fuctioal Aalysis. But ifiite dimesioal spaces eed ot always be completely mysterious; sometimes oe lucks out ad ca watch a couterituitive pheomeo developig i R for large. This might be of use i oe of several ways: perhaps the behavior for large but fiite is already useful, or oe ca deduce a iterestig statemet about lim of somethig, or a peculiarity of ifiite-dimesioal spaces is illumiated. I will describe some curious features of cubes ad balls i R, as. These illustrate a pheomeo called cocetratio of measure. It will tur out that the importat law of large umbers from probability theory is just oe maifestatio of high-dimesioal geometry. Alog the way, we will meet some stadard aalysis techiques. These may be familiar to varyig degrees. I thik it could be useful for you to see how multidimesioal itegrals, liear algebra, estimates, ad asymptotics appear i the aalysis of a cocrete problem. A umber of these matters are relegated to appedices. I the mai part of the expositio, I try to focus (as much as possible) o geometric pheomea; it would be reasoable to read about those first, ad oly c by Herma Flaschka.

2 MATH 57A refer to a appedix for quick clarificatio as eeded. Ultimately, you should also uderstad the appedices.. The Cube.. Volume of the cube. C (s) is the cube cetered at the origi i R with sidelegth s. I.e., C (s) = {(x,..., x ) s x j s for all j}. Its -dimesioal volume is (by defiitio) Vol(C (s)) = (s) (s) (s) = (s). } {{ } times We have a obvious cosequece: Propositio.. As, the volume of C (s) teds to zero if s <, to if s >, ad it is always = for s =. From ow o, C ( ) will be my referece cube, ad I will simply write C. So Vol(C ) = for all. Notice, however, that the poit (,..., ) is a vertex of the cube, ad it has distace / from the origi. So Propositio.. The cube C has diameter, but volume. The mathematics is completely elemetary, but I hope you agree that visualizig such behavior is rather more tricky. It gets worse... Cocetratio of volume. I wat to compare the volume of C to the volume of a subcube, C ( ɛ ), where ɛ (, ) is give. We already kow from Propositio. that the latter volume teds to zero as. Hece the shell betwee the two cubes cotais most of the volume: Propositio.3. For every ɛ (, ), teds to as. Volume of shell = Vol(C C ( ɛ )) I other words, as, there is o such thig as a thi shell of small volume. All the volume of C escapes towards its surface, out of ay prescribed subcube. I order to uderstad, i some measure, how the volume cocetrates at the surface of C, look agai at (.) Vol(C C ( ɛ )) = ( ɛ).

3 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 3 Of course, ( ɛ) because < ɛ <. Now ivoke Lemma A.: lim ( + a ) = e a. This suggests that we should let the ɛ i (.) vary with. Istead of takig a subcube C ( ɛ ) whose sides are i fixed ratio to the side (legth = ) of C, we expad the subcube, ad shrik the shell, as icreases. I this way, the volume trapped i the shrikig shell does have a ozero limit: Propositio.4. For every t >, lim Vol(C C ( t )) = lim ( t ) = e t. [It is uderstood that must be large eough to esure t < ]. How to thik about this? Say you wat to kow what shell cotais half the volume of C (for large, of course). Sice e t =.5 for t = , you kow that the cube of sidelegth has volume about, with the remaiig volume cotaied i the very thi shell (of width.69...) betwee the subcube ad our referece cube, C. Later o, we will look at the cube a bit differetly, ad see that its volume also cocetrates alog the diagoal plae x + + x =. This will be related to probability theory. First, however, we will study the ball; it is more symmetrical tha the cube, ad easier to deal with..3. Surface area of the cube. I dimesios ad, cubes are ot cubes i the everyday sese of the word. The cube C (s) is a lie segmet of legth s. The cube C (s) is a square, sidelegth s. It has four -dimesioal sides, each of which is a copy of C (s). The surface area of C (s) is the total -dimesioal volume, i this case just ordiary legth, of the four sides, to wit 4 (s). C 3 (s) is the usual cube. Its six -dimesioal sides are copies of C (s), ad the surface area of C 3 (s) is the sum of the -dimesioal volumes, or the usual areas, of its sides: 6 (s). The cube C 4 (s) i four dimesios has a umber of 3-dimesioal sides. Let s call them 3-faces. The surface area of C 4 (s) is really the sum of the 3-dimesioal volumes of the 3-faces: 8 (s) 3. To see why this formula is correct, we eed to describe the 3-faces. A 3-face is determied by settig oe of the coordiates equal to its extreme value, ±s, ad allowig the other coordiates to vary i [ s, s].

4 4 MATH 57A For example, {(s, x, x 3, x 4 ) x j s for j =, 3, 4} is a 3-face. It is a copy of the cube C 3 (s) = {(x, x, x 3 ) x j s for j =,, 3}; except the idices have chaged (which is irrelevat). This object has the 3-dimesioal volume (s) 3. There are eight 3-faces, sice ay oe of the four coordiates could be fixed at +s or s, with the other three coordiates allowed to vary i [ s, s]. The surface area of C (s) is the sum of the ( )-dimesioal volumes of its ( )-faces. Area is ot really a good term, but we will use it to distiguish betwee the -dimesioal volume of the solid -dimesioal object, ad the ( )-dimesioal volume of the boudary of the object. Later o, the same covetio will apply to the ball. Exercise.. How may ( )-faces does C (s) have? How may ( k)-dimesioal faces, for k? (A -face is a vertex, ad the -face, by covetio, is the whole solid cube). Exercise.. For each s (, ), compare the behavior of the volume of C (s) to the behavior of its surface area, as. The case s = should strike you as beig couterituitive. Exercise.3. Fix ɛ (, s). Let S ɛ be the slice S ɛ = C (s) {(x,..., x, z) z < ɛ}, i.e. a sort of equatorial slice of the cube. How does the ratio behave as? Vol(S ɛ ) Vol(C (s)) 3. Volume of the Ball The material about cocetratio of volume comes (at times verbatim) from P. Lévy, Problèmes cocrets d aalyse foctioelle, Gauthier- Villars (95), p.9 ff. See also pp. 6-6 of the Math 57 otes for the computatio of the volume of a ball i R. somethig like ( )-dimesioal Lebesgue measure might be preferred by cogosceti

5 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 5 x d(si ) cos si IR Figure. Volume of ball by slices 3.. Volume of the ball, part. We write B (R) for the solid ball, radius R, cetered at the origi i R. The term ball meas surface plus iterior. The surface itself is called the sphere, deoted by S (R). The superscript ( ) sigifies that the sphere i R has dimesio ( ). Thus B (R) = {(x,..., x ) R x + x R}, S (R) = {(x,..., x ) R x + x = R}. Example 3.. To be clear about dimesios, cosider B 4 () = {(x, x, x 3, x 4 ) x + x + x 3 + x 4 } R 4. This is 4-dimesioal, i the sese that as log as x +x +x 3 +x 4 <, all four coordiates ca be varied idepedetly. However, S 3 () = {(x, x, x 3, x 4 ) x + x + x 3 + x 4 = } R 4. This is 3-dimesioal, i the sese that oce three of the x j are fixed, the remaiig oe is largely determied (up to ± sig, i this example). There are better defiitios, but this should covey the idea. It is rather tricky to picture S 3, sice we ca t go ito four dimesios, but there are ways. I ay case, this sectio deals with the ball oly.

6 6 MATH 57A Now let K be the volume of the uit ball B () i R. The the volume of B (R) is K R. Let us calculate K (refer to Figure ). The circle represets the ball of radius. The horizotal axis is R, the set of ( )-tuples (x,..., x ). The vertical axis is the last directio, x. If we were computig the volume of the ball i R 3 (i.e., = 3), we would add up the volumes of circular slices of radius cos θ ad thickess d(si θ): K 3 = π π π cos θ d(si θ). Now this itegrad π cos θ is the two-dimesioal volume of the twodimesioal ball (=disk) of radius cos θ. Note that it has the form volume of two-dimesioal uit ball radius = π cos θ. Usig symmetry i θ, we may thus write π K 3 = K cos 3 θ dθ. The patter cotiues for > 3. I claim that x = si θ, represeted by a horizotal lie segmet i the figure, itersects B () i a ( )- dimesioal ball of radius cos θ. Ideed, the itersectio is defied by (3.) x + + x + si θ = x + + x si θ = (cos θ), ad this is B (cos θ). So, istead of a very thi slice with a twodimesioal disk as cross-sectio, we ow have a very thi (d(si θ)) slice whose cross-sectio is the ball of radius cos θ i dimesio ( ). Its volume is K cos θ d(si θ). Thus, π (3.) K = K cos θ dθ = I K, where (3.3) I def = π cos θ dθ. To make further progress towards a useful formula for K, we eed to uderstad the itegral I.

7 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES The itegral I. Here we collect iformatio about I. We retur to the volume K i the ext subsectio. Itegratio by parts i (3.3) gives (for > ) Hece I = [si θ cos θ] π + ( ) = ( ) π = ( )(I I ). π (cos θ cos θ) dθ cos θ si θ dθ (3.4) I = I. Sice I = π ad I =, we ca solve (3.4) recursively. Startig with I, we fid I, I 4,..., ad from I we get I 3, I 5,.... The patter is easy to ascertai: I p = π 3 5 (p ) (3.5), 4 6 (p) (3.6) I p+ = 4 6 (p) 3 5 (p + ). Remark 3.. Stadard otatio: (p + ) = (p + )!!. We will eed to kow how I behaves as. Sice the itegrad cos θ decreases with for every θ, oe expects that I. This is true, but we wat to kow how fast the decrease is. The ext lemma provides the aswer. We use the otatio a b to sigify that lim a b = ; this would be a good time to glace at Appedix B. Lemma 3.. I π. Proof. From (3.5) ad (3.6), oe sees that for all positive, (3.7) I I = π. (Aother proof: I = ( )I implies (3.8) I I = ( )I I ; the writig (3.8) for with replaced by,,..., we get the strig of equalities I I = ( )I I = ( )I I 3 = = I I = π.) Here is the idea of the rest of the proof. First we ote that by (3.4), I I. Next we show that I is trapped betwee I ad I.

8 8 MATH 57A It will follow that I I, so that (3.7) gives I π. Takig the square root will establish the Lemma. Let s do it more carefully. Because cos θ, we have for θ (, π ), ad thus I Divide by I ad use (3.7): cos θ < cos θ < cos θ < I < I. < I I <. Sice the right side has limit, it follows that Now multiply (3.7) by I I I lim =. I ad rearrage: ( ) I = π/() I I. The right side has limit =, hece so does the left side. Now take the square root. Remark 3.. Lemma 3. has a remarkable cosequece. = p +, it says that or i loghad, lim p (p + ) I p+ = π, 4 (p) (3.9) lim p 3 5 (p ) p + = π. Applied to There is a dot-dot-dot otatio that is used for ifiite products, aalogous to the usual otatio for ifiite series (here... stads for a certai limit of partial products, aalogous to the familiar partial sums); oe would write π = This is called Wallis product. J. Wallis, Arithmetica ifiitorum, Oxford 656 (!). The product ca be recast as Π (4 /(4 )). By the way, Wallis was a codebreaker for Oliver Cromwell, ad later chaplai to Charles II after the restoratio

9 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 9 Let us ow look at π cos θ dθ more closely. The itegrad is decreasig i θ, ad for each θ (where its value is = ), it decreases to zero with. So the mai cotributio to the itegral should come from a eighborhood of θ =. Ideed, if α > ad θ (α, π ], the cos θ < cos α ad so π π cos θ dθ < cos α dθ = ( π α) cos α. α α Now cos α approaches zero geometrically, but we saw i Lemma 3. that I approaches zero oly like. We are i the situatio of the example i Appedix B ( geometric decrease beats power growth ), which gives (cos α) : We obtai: I π α cos θ dθ. Propositio 3.. For every α (, π ], I α cos θ dθ. If, however, the limit of itegratio α is allowed to decrease as icreases, oe ca trap a proper fractio of the the total value of I i the shrikig iterval of itegratio. Cosider β From (A-6) i Appedix A, ad we just saw that Thus β (3.) lim I cos θ dθ = β cos ( t ) dt β I π. cos θ dθ = cos ( t ) dt. β e t dt, β e t π Remark 3.3. By the time we get to (3.), we o loger eed β (, π/). That restrictio was used i the earlier observatio that the itegral over ay fiite subiterval [, α] [, π/] was asymptotically dt.

10 MATH 57A equivalet to I. Oce we go to the shrikig upper limit α = β/, the this upper limit will fall ito (, π/] whe is large eough, o matter the choice of β Volume of the ball, part. We had obtaied the relatio (3.), K = π K cos θ dθ = I K. Replace by ( ) to get K = I K, ad thus K = 4I I K. Now recallig (3.7) from the precedig subsectio, we obtai a recursio relatio from which K ca be determied: (3.) K = π K. Kowig that K = π ad K 3 = 4π, we fid K 3 4, K 6,... ad K 5, K 7,..., ad recogize the patter: (3.) (3.3) π p K p = p! (π) p K p+ = 3 5 (p + ). This formula has a amazig cosequece: Propositio 3.. For every R >, the volume of the ball of radius R i R teds to zero as. Proof. If = p is eve, for example, the volume will be K p R p = (πr ) p. p! Thus K p+ R p+ (πr ) = (p + )(p + ) K pr p, showig that the volume decreases rapidly, oce p becomes sufficietly large. (If you otice that the K p R p happe to be the coefficiets i the Taylor series represetatio of e x whe x = πr, this argumet should rig a bell). The proof for odd dimesios is similar. As was see i Propositio., the depedece of the volume of the cube C (s) o s ad is quite differet. There is aother curious relatio betwee cubes ad balls that should be oticed. The distace betwee the origi ad the vertex (s,..., s) of C (s) is s. Therefore, the radius (=s ) of the smallest ball

11 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES cotaiig C (s) teds to with. Furthermore, if s >, the Vol(B (s )) (sice the volume of the iscribed cube teds to ifiity). O the other had, the largest ball cotaied i C (s) will have radius s, which is idepedet of. Eve if Vol(C (s)), the volume of this iscribed ball B (s) teds to zero. It is temptig to extrapolate to ifiite dimesios. I R (whatever that might mea), a cube is ot cotaied i ay ball of fiite radius. The cube with sides of legth = ad volume = cotais the uit ball, which has volume =. Well, this last statemet is mostly osese, but somewhere i it there is a little truth Asymptotic behavior of the volume. The formula for K ivolves factorials, which are approximated by Stirlig s formula (C-), Propositio 3.3. k! πk k k e k. Vol(B (R )) (πe) π R. Proof. Apply Stirlig s formula, first i eve dimesios. From (3.), K p π p πp pp e p ; if we set p = / ad do some rearragig (exercise), we get (3.4) K (πe) π( ). Multiplyig this by (radius) = ( R), we get the desired formula. The case of odd dimesios is somethig of a mess, ad there is a subtle poit. Begi by multiplyig top ad bottom of the expressio (3.3) for K p+ by 4 6 (p) = p p!, ad also multiply from the begiig by the factor ( ) that will come from ( R), to get (recall that = p + ) (π) p p p! p+ (p + ). (p + )! Next, substitute for the two factorials from Stirlig s formula. terms are ordered so that certai combiatios are easier to see. πp (π) p p e p p p p+ (p + ) π(p + ) e p. (p + ) p+ The

12 MATH 57A First some obvious algebraic simplificatios i the last expressio: p (π) p p e p+ p p (p + ) p + (p + ) p. Moreover, p p (p + ) = ( ) p p p p p + = p ( +, p )p ad isertig this leads to p (3.5) K p+ (π) p e p+ p + ( + (p + ). )p p Now we do asymptotic simplificatios as p. The subtle poit arises here. Let RHS stad for the right side of (3.5). So far we kow that (3.6) lim p K p+ RHS =. If we replace ay part of RHS by a asymptotically equivalet expressio, obtaiig a modified expressio RHS, say, we will still have For example, We ca the modify (3.6): K p+ lim p RHS =. lim p K p+ lim p RHS p p+ =. p p+ =. This gives a ew deomiator, ad we get lim p K p+ (π) p e p+ (+ p )p (p + ) which we would write more briefly as =, (3.7) K p+ (π) p e p+ ( + (p + ). )p p A compariso of (3.5) ad (3.7) suggests that we have allowed p to ted to i some places, but ot i others. Is t this uethical? No it is ot, as log as we are replacig oe expressio whose ratio with K p+ teds to by aother expressio with the same property.

13 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 3 This same reasoig further allows us to substitute e for ( + p )p. By ow, (3.5) has bee simplified to (π) p e p+ (3.8). (p + ) ad it is a easy maipulatio to tur this ito the desired formula i terms of = p +. Corollary 3.. lim Vol(B (R )) = {, if R πe, if R > πe. Proof. Write the asymptotic expressio for the volume as ( πer ) π, ad recall that whe πer >, the umerator grows much faster tha the deomiator. The case is trivial. This is a little more like the behavior of the volume of the cube, especially if you thik ot about the side legth s of C (s), which remais fixed, but about the legth s of its diagoal: Cube: diameter = costat, Ball: radius = costat. The behavior of the volume is determied by the value of the proportioality costat. Still, there is o way to get a ozero, o-ifiite limitig volume for the ball Volume cocetratio ear the surface. Cosider the two cocetric balls B () ad B ( ɛ). The ratio of volumes is Vol(B ( ɛ)) Vol(B ()) = K ( ɛ) K = ( ɛ). For every ɛ, this ratio teds to zero as, which meas that every spherical shell, o matter how thi, will cotai essetially the whole volume of B (). Of course, it should be remembered that lim Vol(B ()) =, so the shell also has a small volume, it is just a sigificat fractio of the whole.

14 4 MATH 57A To see how the volume accumulates at the surface, we agai let ɛ deped o, as we did for the cube. Choosig ɛ = t, we fid that Vol(B ( t )) Vol(B ()) = K ( t ) K = ( t ) e t. The iterpretatio is exactly as for the cube (subsectio.) Volume cocetratio ear the equator for B (R). Recall that Vol(B (R)) for every R. We ask: what fractio of this vaishig (as ) volume accumulates ear the equator of the ball? Let θ < < θ, ad cosider (3.9) R θ θ cos θ dθ R π π = ( θ ) cos θ dθ + cos θ dθ. I θ The umerator is the volume of the slice bouded by the hyperplaes 3 x = R si θ, x = R si θ (see Figure ). Sice the factor R has cacelled, we will temporarily set R = ad work with the uit ball; later, R will be re-iserted by had. Accordig to Propositio 3., the two itegrals o the right side of (3.9) are each asymptotically equivalet to I. Hece: Propositio 3.4. For every θ [ π, ) ad θ (, π ], the fractio of the volume of B (R) cotaied i the equatorial slice betwee the hyperplaes x = si θ ad x = si θ teds to as. If, however, θ > (θ < θ ), the the fractio of volume i the slice teds to zero. As before, to trap a proper fractio of the volume, we must allow the limits of itegratio to approach as. The results of Subsectio 3. suggest that we take θ = β ad θ = β, β < β. The sigs of the β s will ot matter ow, oly the fact that β j at a cleverly chose rate. Icidetally, sice si u u for u, the boudig hyperplaes x = si( β j ) are essetially x = β j as gets large. From equatio (3.), we ow obtai 3 a hyperplae i R is a -dimesioal set defied by a sigle liear equatio i x,..., x. If this set does ot cotai the origi, it is sometimes called a affie hyperplae, but I wo t make that distictio. Hyper meas oe dimesio dow, i this cotext. The set defied by a sigle, geerally oliear, equatio f(x,..., x ) = is a hypersurface. The sphere S (R) is a hypersurface i R.

15 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 5 Propositio 3.5. Let β < β. The fractio of the volume of B () cotaied i the slice bouded by the hyperplaes x = si( β j ), j =,, teds to (3.) π β β e t Remark 3.4. Propositio 3.5 remais true if oe asks about the volume betwee the hyperplaes x = β j, j =,. That volume differs from the volume i the propositio by si ( β ) cos θ dθ β ad aother similar itegral, but the limits of itegratio approach each other as ad the itegral teds to zero. The details are omitted. Remark 3.5. If you kow some probability theory, you will have oticed that (3.) represets a area uder the stadard ormal curve. It is a ofte used fact that the value of (3.) for β =.96, β = +.96 is approximately.95. I other words, for large all but about 5% of the volume of B (R) will be foud i the very thi slice betwee x = ±.96R. Remark 3.6. By rotatioal symmetry, Propositio 3.5 will hold for every equator, ot just the oe perpedicular to the x -directio. Thik about that for a miute. Next, icorporate this i your thoughts: the volume also cocetrates ear the surface of the ball Volume cocetratio ear the equator for B (R ). The results of the last subsectio ca be viewed from a slightly differet agle, which is importat eough to be emphasized i its ow little subsectio. Istead of fixig R, ad lookig at the fractio of volume cotaied i the slice betwee 4 x = β ad x = β, oe ca let the radius of the ball become ifiite at the rate R ad look at the slice betwee x = β ad x = β. Keep i mid that Vol(B ( R )) or, so we may be talkig about a fractio of a tiy volume or of a huge volume. dt. 4 it is simpler to use β/ istead of si(β/ ); the two are asymptotically equivalet

16 6 MATH 57A Propositio 3.6. Let β < β. The fractio of the volume of B (R ) cotaied i the slice bouded by the hyperplaes x = β j, j =,, teds to (3.) π β Exercise 3.. Prove this propositio. β e t dt. 4. Surface area of the ball 4.. A formula for surface area. Recall that the surface of a ball is a sphere. We speak of the area of a sphere simply to distiguish that quatity from the volume of the whole ball. But keep i mid (Example 3.) that the sphere S 3 (R) is itself 3-dimesioal (sittig i the 4-dimesioal R 4 ). What we refer to as its area is actually its 3-dimesioal volume. Likewise, the area of S (R) R is uderstood to mea its ( )-dimesioal volume. The surface area of B (R) is easily expressed i terms of Vol(B (R)). Cosider the -dimesioal volume of the spherical shell betwee the balls of radii R + δr ad R: Vol(B (R + δr)) Vol(B (R)) = K [(R + δr) R ] = K [R δr + ], with deotig terms i (δr) ad higher. Thik of this volume as beig approximately (4.) volume of shell = area of S (R) thickess δr of shell, i.e. volume of shell = K R δr +. (4.) becomes more accurate as δr. Hece the area of S is d (4.) dr Vol(B (R)) = K R. This type of argumet would ideed recover a formula which you kow from calculus, for the area of a regio D o a surface (i R 3 ) give by z = f(x, y): + fx + fy dx dy. D But we will eed to get at the area of the sphere by a differet, seemigly more rigorous approach ayway.

17 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 7 x d si cos IR Figure. Surface area by slices Let L deote the surface area of the uit ball B () i R. We kow from geeral priciples that the ball B (R) will have surface area L R ; this will be used shortly. Now refer to Figure, which agai represets the uit ball B () i R. (If you are at all ueasy with the use of i what follows, work through the steps usig = 3). Recall from (3.) that the lie segmet x = si θ represets the ball B (cos θ); as just remarked, it will have surface area L (cos θ). This is ( )-dimesioal volume. The outside arc of the shaded area is the surface of a cylider put together as follows: the base is B (cos θ) ad the edge (surface) of the base is S (cos θ); the height is dθ. For small dθ, the side of the cylider is early perpedicular to the base, ad the the outside arc of the shaded regio represets a cotributio circumferece height = L (cos θ) dθ to the total surface area. Therefore (4.3) L = π π L cos θ dθ = I L. To obtai a formula for L, we ca proceed i three ways. First, i the formula L = K, which is (4.) at R =, use the kow expressios for K. Secod, i (4.3), start with = 3 ad L 3 = L =

18 8 MATH 57A Figure 3. Volume vs. surface area π, ad use the kow expressios for I. Third, i (4.3), replace L by I 3 L 3 (which is (4.3) after the replacemet ( ) ( )), to get L = 4I I 3 L 3 = π L 3 (last step by (3.7)). The use L = π ad L = 4π to get started o a recursive determiatio of L. Presumably, all these methods lead to the same aswer: Propositio 4.. L p = 4π (π)p (p )!! L p = π πp p!. Remark. It is worth uderstadig exactly why cos θ becomes cos θ, ad ot cos θ, which would be aalogous to the chage i the expoet of R. For the volume, we had which was the multiplied by volume of base of slice (radius) thickess of slice = d(si θ) = cos θ dθ. For the surface area, it was surface area of base of slice (radius) (here the power does just drop by ), but this was oly multiplied by the arclegth dθ. So the differece is the bit of rectagle that sticks out beyod the circular arc i Figure 3. Exercise 4.. Compare the behavior of the volume ad surface area of B (R ), for all R >. Oe value of R should be peculiar.

19 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES Surface area cocetratio ear the equator. The surface area of B (R) cocetrates ear the (or ay) equator. The calculatio has already bee doe. Look at (3.9). If R is replaced by R, ad cos θ is replaced by cos θ, (3.9) represets the fractio of surface area of B (R) cotaied betwee the hyperplaes x = si θ j, j =,. Everythig we kow about the limits remais true, word for word. I will ot restate the results, you do it: Exercise 4.. State precisely the propositio about surface area cocetratio ear the equator. 5. The weak law of large umbers I this sectio, cocetratio of volume for the cube about its equator is show to be a expressio of oe of the basic theorems of probability theory. I will use several techical probability defiitios i a ituitive way; otherwise, there would be too may digressios. 5.. The probability settig. Let C = C agai be the iterval ( ) = [, ], ad cosider the experimet of choosig a umber at radom from this iterval. All results are to be cosidered equally likely. More precisely, the probability that a radomly chose umber x falls ito a subiterval [a, b] [, ] is defied to be its legth, b a. We write P (x [a, b]) = b a. (So P (x [, ]) = ). I a setup like this, where the possible choices form a cotiuum, the probability that a radom x will be exactly equal to a give umber is zero. For example, P (x = ) =. A justificatio might be that you would ever kow x to all its decimal places. More mathematically, this is the oly possible logical cosequece of our startig defiitio of the probability: P (x [, ]) = =. Now we go to C (uit side legth, remember), ad defie the probability that a radomly chose poit lies i a regio D C to be the volume, Vol(D). Agai, the whole cube has probability. Oe ca imagie pickig x = (x,..., x ) as a whole, e. g. by throwig a dart ito the -dimesioal cube. Or ad this is what I will do oe ca build up the radom x oe coordiate at a time. First choose x C at radom. The, idepedetly of this first choice, pick x C. Cotiue. The probability that a x built i such a way lies i D is agai Vol(D). This is ot easy to prove with full mathematical rigor, oe essetially eeds the theory of Lebesgue measure. I will illustrate for the case = ad D = a rectagle. The key techical otio, which caot be avoided, is idepedece.

20 MATH 57A Summary. A experimet with a umber (maybe ifiite) of possible outcomes is performed. A evet A is a subset of the set of possible outcomes. A probability measure assigs umbers betwee ad to evets. The iterpretatio of P (A) = p is that if the experimet is repeated a large umber N times, the about pn of the results will fit the criteria defiig A. (Every sigle statemet I just made requires clarificatio ad much more mathematical precisio.) Example 5.. Suppose a 6-sided die is tossed 6, times. Let A =, 4, or 6 shows, ad B =, 3, 4, or 6 shows. Assumig that each of the six outcomes,, 3, 4, 5, 6 is equally likely, we have P (A) = ad P (B) = 3. This meas, ituitively, that A should happe o about half, or 3, of the tosses. Of these 3,, about oe third each will result i i, 4, 6, so B happes o /3 or, tosses. Tosses resultig i or 3 are o loger possible, sice we are restrictig attetio to outcomes where A has occurred. I other words, the probability of B was 3 at the outset, ad it was still after the results were restricted to the outcomes, 4, 6 i A. Oe says: 3 (5.) the probability of B give A = the probability of B. We caot make a better predictio about the occurrece of B amogst the outcomes satisfyig A tha we could about the occurrece of B amogst all possible outcomes. Example 5.. Suppose ow B is,3, or 5 shows. The oe of the 3, tosses fittig the descriptio of A will fit the descriptio of B. Thus: the probability of B give A =. If we kow that A has occurred, we kow with certaity that B has ot occurred. We cosider the A ad B i Example 5. to be idepedet. Example 5. illustrates a extreme case of o-idepedece. Aother extreme case would be B =,, 4, or 6 shows. If we kow that A has happeed, it is certai that B has also happeed, or the probability of B give A =. The techical meaig of idepedece is a rephrasig of (5.): (5.) P (A ad B both happe ) = P (A)P (B). To uderstad this, rewrite it as (if P (A) ) (5.3) P (A ad B both happe ) P (A) = P (B).

21 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES The left side is the fractio of the outcomes satisfyig A for which B also happes. I the settig of Example 5., it would be, 3, = 3 6, 6, = 3 = P (B). Example 5.3. We will pick a poit x C at radom, by choosig its two coordiates idepedetly. Let A be the evet x [a, b ], ad let A be the evet x [a, b ]. We have P (A ) = b a, P (A ) = b a. The choices of x ad x are to be made idepedetly. This meas that we require that P (x [a, b ] ad x [a, b ]) = (b a )(b a ). But this is the same as sayig that P (x = (x, x ) rectagle with base [a, b ] ad height [a, b ]) = Vol(rectagle). The geeral statemet P (x D) = Vol(D) is deduced by approximatig the regio D by a uio of small rectagles (compare the defiitio of a multiple itegral as limit of Riema sums). Laws of large umbers give precise meaig to statemets that somethig should happe o average. For example, if we choose, radomly ad idepedetly, a large umber N of poits i C = [, ], say a, a,..., a N, ay positive a j should sooer or later be early balaced by a egative oe, ad the average a + a + + a N (5.4) N should be close to zero. The larger N is, the better the law of averages should apply, ad this ratio should be closer to zero. Now, as just explaied, these idepedetly chose a j ca be thought of as the coordiates of a radomly picked poit a = (a,..., a N ) i the high-dimesioal cube C N. Oe versio of the law of averages says that the volume of the regio i C N where (5.4) is ot close to zero becomes egligible as N. I other words, we have cocetratio of volume agai: ( Vol {a = (a,..., a N ) C N a + a + + a N N ) is small}. I ow tur to the geometric problem, ad revert to my usual otatio (x,..., x ) C. The iterest is i cocetratio of volume about the plae Π : x + + x =. Notice that this plae has ormal (,,..., ), ad that the lie alog the ormal is i fact the mai

22 MATH 57A diagoal of the cube, passig through the corer (,,..., ). The distace to the corer is. We wat to look at slices bouded by plaes Π ±η : x +... = ±η; possibly η will deped o we shall see. The hoest approach would be to fid a ice, more or less explicit, formula for the volume of such a equatorial slice, as there was for a ball, ad the to apply the geometric fact of cocetratio of volume to obtai a probabilistic applicatio. However, I cofused myself tryig to fid a volume formula; there are too may corers whe you slice the cube by a plae Π η, ad I gave up. The less satisfyig approach is to prove the probabilistic result as it is doe i probability texts (usig a simple iequality, see 57 Notes, p. 386), ad to reiterpret it i geometric laguage. This is what I have to do. 5.. The weak law of large umbers. For x = (x,..., x ) C, set S (x) = x + + x ; this is a fuctio whose domai is the cube. I will sometimes omit the argumet x. Further, give ɛ >, defie the equatorial slice (5.5) S ɛ = {x C S (x) < ɛ}. Theorem 5. (Weak law of large umers). For every ɛ >, Equivaletly, (5.6) lim Vol ( lim Vol(S ɛ) =. {x C S ) (x) > ɛ} =. Remark 5.. The weak law is really a much more geeral theorem. I have stated it i the cotext of our cube example. But the x j might also be chose at radom from the two-elemet set {, }, with P ( ) = P () =. These two umbers could sigify tails ad heads i a toss of a fair coi. We would the be workig with a discrete cube, {, }. Theorem 5. gives oe precise meaig to the law of averages for coi tosses: there are, o average, about as may heads as there are tails. Remark 5.. If there is a weak law, there should be a strog law, ad there is. It is set i the limitig cube, C, where ow x = (x, x,...) is a -tuple. It says that those x for which x + + x N (5.7) lim = N N

23 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 3 occupy total volume i C. Evidetly, some more theory is eeded here, to give meaig to the otio of volume i C, ad to clarify the ature of possible sets of zero volume (where the limit (5.7) fails to hold). The proof of Theorem 5. is based o a useful iequality, which is stated here for the cube C, although ay regio of itegratio could be used. For typig coveiece, I use a sigle itegral sig to stad for a -fold multiple itegral, ad I abbreviate dx dx = d x: dx dx d x. } {{ } C C Lemma 5. (Chebyshev s iequality). Let F : C R be a fuctio for which σ def = F (x) d x is fiite. For δ >, let The A δ = {x C F (x) > δ}. (5.8) Vol(A δ ) < σ δ. Proof. (of lemma) Sice A δ C, F (x) d x F (x) d x = σ. A δ C For x A δ we have (δ) < (F (x)), so δ d x < F (x) d x. A δ A δ The left side is just δ A δ d x = δ Vol(A δ ). Combie this strig of iequalities to get as desired. δ Vol(A δ ) < σ, Proof. (of Theorem) We apply Chebyshev s iequality to F = S. First, we ote that (S ) = x j + x i x j. j= i= j= j i

24 4 MATH 57A The itegral over C of the double sum vaishes, sice i the iterated multiple itegral, x i dx i =. A typical cotributio from the simple sum is ( ) x = x dx dx... dx = C. Thus, σ =. Chebyshev s iequality ow gives ( ) Vol {x S (x) > δ} Set δ = ɛ i this relatio, ad you get ( (5.9) Vol {x S ) (x) > ɛ} < δ. < ɛ. This teds to zero as, ad the theorem is proved Coectio with cocetratio of volume. We have see that the volume of the regio C where S < ɛ teds to as. The geometric meaig of this fact still eeds a little more clarificatio. To this ed, we switch to a coordiate system i which oe directio is perpedicular to the plaes Π η : x + +x = η. This ew directio is aalogous to the x -directio i our discussio of the ball. Let e,..., e be the stadard basis, Thus, e j = (,...,, }{{},,..., ). j th place x = x e + + x e. Now let f be the uit vector (,..., ) = (e + + e ) i the directio ormal to Π. The orthogoal complemet to f is of course the ( )-dimesioal vector subspace Π = {x R x + + x = }.

25 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 5 The process of Gram-Schmidt orthogoaliztio i liear algebra guaratees that we ca start with the uit vector f, ad costruct orthoormal vectors f,..., f which, together with f, provide a ew orthoormal basis of R. (These f j are far from uique; certaily oe possible set ca be writte dow, but that would be more cofusig tha helpful). The poit is that a give x i the cube ow has two coordiate expressios: x e + + x e = y f + + y f + y f. Take ier products of both sides with f, ad remember that the f j are orthoormal. You fid (5.) y = (x + + x ) = S. Now go back to (5.9). Isertig (5.), we covert it ito ( Vol {y y > ɛ ) } < ɛ. Fially, otice that the corer of the cube, where all x j =, correspods to y =. We may therefore phrase the weak law of large umbers i more geometric laguage, as follows. Propositio 5.. The diagoal of C has legth. The fractio of the (uit) volume of C cotaied i the slice S ɛ (see (5.5)) approaches as. I other words, a slice S ɛ whose width is a arbitrarily small fractio of the legth of the diagoal of C will evetually eclose almost all the volume of C. Remark 5.3. Recall that the volume of C also cocetrates ear the surface! Exercise 5.. Verify that oe possible choice for the ew basis vectors f j is, for j =,..., j f j = (,...,,,,..., ). j(j + ) j(j + ) j(j + ) } {{ } j terms

26 6 MATH 57A A. appedix: compoud iterest I the mai part of these otes, we eed two facts. First, equatio (A-) below, which ofte arises i a discussio of compoud iterest i begiig calculus. It is fairly easy to prove. Secod, the rather more subtle formula (A-) β lim cos ( t ) dt = β lim cos ( t β ) dt = e t which is a cosequece of (A-) ad some other stuff. The first subsectio explais the ideas, ad everythig afterwards fills i the techical mathematical details. There are lots of them, because they will reappear durig the Math 57 year, ad you might as well get a brief itroductio. The details could well detract from the importat issues o first readig. Study the earlier material, ad the go through the details i the appedices later (but do go through them). A.. The ideas. The first result should be familiar. Lemma A.. For every a R, (A-) lim ( + a ) = e a. I will actually require a cosequece of a refiemet of Lemma A.. This geeralizatio of Lemma A. says, i a mathematically precise way, that if b, the we still have (A-3) lim ( + a + b ) = e a, ad that the approach to the limit is equally fast for all a i ay fixed fiite iterval [ A, A]. (Techical term: the covergece is uiform). The relevat applicatio is this uexpected corollary: dt, Corollary A.. (A-4) lim cos ( t ) = e t. Covergece is uiform o fiite itervals, i this sese: give β > ad ɛ >, there is a iteger N > depedig o β ad ɛ but ot o t, such that > N implies (A-5) cos ( t ) e t < ɛ for all t [ β, β].

27 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 7 Figure 4. Illustratio of (A-4) Shelve that defiitio for the momet, ad let me reveal why (A-4) should be true ad why I wat this uiform covergece. From the Taylor series for cos u we have cos u = u +. Suppose all the terms are just ot there. The by (A-), (cos( t )) = ( t ) which has limit e t /! But the are there, i the guise of the correctio term i Taylor s formula. So oe ecouters the more complicated limit (A-3). I will the wat to coclude that (A-6) β lim cos ( t ) dt = β lim cos ( t β ) dt = e t This seems plausible, give (A-4), but it is ot always true that the limit of the itegrals of a sequece of fuctios is equal to the itegral of the limit fuctio. I other words, equality = may fail for some sequeces of fuctios. It helps to look at some graphs. dt.

28 8 MATH 57A y=^*x^*(-x), =,,3,4, Figure 5. f, but f Figure 4 shows the graphs of cos (t/ ) for =, 4, 9, ad the graph of exp( t /) (it is the outermost curve). The cosie powers get very close to each other, ad to the expoetial, over the whole iterval [ π, π]. Take a arbitrarily arrow strip alog the curve exp( t /). All the cos (t/ ), from some o, will fit ito that strip 5. This is the meaig of f f uiformly. Pick ay arrow strip about the graph of the limit fuctio f. The the graphs of f fit ito that strip, from some N owards. Clearly, the areas uder the cosie curves approach the area uder the expoetial curve. This is a obvious, ad importat, cosequece of uiform covergece. Figure 5 shows the graphs of the fuctios f (x) = x ( x), defied for x. Clearly, f () = f () = for all, ad for each idividual x (, ), the geometric decay beats power growth priciple implies that f (x) (see the begiig of Appedix B, if you are ot sure what I mea). So f (x) for every x [, ]. If the covergece f were uiform o the closed iterval [, ], the by defiitio of uiform covergece you could pick a arbitrarily arrow strip about y =, say y < ɛ, ad all f, from some o, would fit ito that strip. This does ot happe here. f (x) at differet rates for differet x. 5 the coveiet strip has the form {(t, y) y exp( t /) < ɛ}. Compare this with equatio (A-5).

29 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 9 This affects the behavior of the itegrals f. We have (x x + ) dx = ( + + ) = ( + )( + ). Agai: eve though f (x) for every idividual x, the itegrals of the f do ot approach zero. A area of size close to migrates to the right. The graph becomes very high over a very arrow iterval, ad the itegrals remai fiite eve though f becomes very small over the rest of [, ]. We express this by sayig that the idetity (A-7) lim f (x) dx = lim f (x) dx does ot hold. But if you look back at (A-4) ad (A-), you see that the property (A-7) is the key to the example. Just to cofuse matters: iterchage of limit ad itegral may be valid eve for sequeces {f } that do ot coverge uiformly. Take f (x) = x ( x), x. As for x ( x), oe has f (x) for each x, i.e. lim f (x) =. Because f (x) dx = ( + )( + ), it is true that lim f (x) dx = = lim f (x) dx. However, covergece is ot uiform. You should check this by drawig graphs of f for a few values of. A more mathematical way to see it is to fid the maximum value of f. It is (/( + )) +, assumed at x = /( + ). Curiously eough, the maximum value approaches exp( ). No matter how large you take, there will always be a bump i the graph that sticks up to about.36, prevetig the graph from fittig ito a arrow strip about the limit fuctio, f(x). Check this. A.. Sequeces of umbers. First, a review of some simple, ad likely familiar, defiitios ad facts about limits of sequeces of umbers. Defiitio A.. A sequece {c } = coverges to c as if for every ɛ > there is a iteger N(ɛ) > (i.e., depedig i geeral o ɛ) such that > N(ɛ) implies c c < ɛ.

30 3 MATH 57A We will wat to plug sequeces ito cotiuous fuctios, ad use the followig property (sometimes take as defiitio of cotiuity, because it is easy to picture). (A-8) If F is cotiuous, the c c = F (c ) F (c). We will prove later that (A-9) l( + a ) a. Takig F to be the expoetial fuctio exp, c = l( + a ), ad c = a, we the obtai from (A-8) ad (A-9) that (A-) ( + a ) = exp[ l( + a )] ea, which is oe of our goals. While (A-8) is a very practical defiitio of cotiuity, the official oe ca be adapted to more geeral settigs; we will do later i the course. Defiitio A.. A fuctio F : R R is cotiuous at c R, if for every ɛ > there is a δ(ɛ) (i.e., depedig i geeral o ɛ), such that x c < δ(ɛ) implies F (x) F (c) < ɛ. If f satisfies this coditio, oe ca deduce the property (A-8). If you are ot familiar with this style of proof, read it. Claim: Suppose that F is cotiuous at c, i the sese of Defiitio A.. If c c, the F (c ) F (c). We eed to show that give ɛ >, there is a N(ɛ) >, such that > N(ɛ) implies F (c ) F (c) < ɛ. So let ɛ > be give. Accordig to Defiitio A., we are guarateed a δ(ɛ) so that x c < δ(ɛ) implies F (x) F (c) < ɛ. Now use Defiitio A. of covergece, but substitute our δ for the ɛ i the defiitio: There is a N(δ) > such that > N(δ) implies c c < δ; this N depeds o δ, ad through δ also o ɛ. The implicatios > N(δ(ɛ)) = c c < δ = F (c ) F (c) < ɛ establish the claim.

31 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 3 A.3. Sequeces of fuctios. So much for sequeces of umbers, ad how they behave whe plugged ito a cotiuous fuctio; however, we will eed to work with sequeces of fuctios. Let us retur to uiform covergece. The picture oe should have i mid is, as explaied above: (A-) the graphs of f are close to the graph of lim f over the whole iterval Defiitio A.3. A sequece of fuctios {f } = coverges uiformly to the fuctio f o the iterval [ A, A], if for each ɛ > there is a N such that > N implies f (a) f(a) < ɛ, for all a [ A, A]. (A = could occur, or the iterval might be replaced by oe of the more geeral form [α, β]). Example A.. Remember that the fuctios f (x) = x ( x) do ot approach the idetically zero fuctio uiformly o [, ]. It was easy to see that the verbal descriptio (A-) is violated, but perhaps it is less clear how to deal with the ɛ s ad N s. If we wat to show that the criterio of the defiitio is ot satisfied, we must show that its egatio holds: there exists ɛ >, such that for every N >, there exist > N ad a [, ] for which f (a) > ɛ. aside: Sice egatig more ivolved statemets could, later i the course, be more tricky, a brief elaboratio might be warrated. The defiitio is built from for all ad there exists statemets, ested i a Russia-doll sort of way. ) for all ɛ >, the followig is true: ) there exists a N for which the followig is true: 3) for all > N the followig is true: 4) for all a [, ], the followig is true: 5) f (a) f(a) < ɛ. The egatio of X is true for all is X is false for at least oe, ad the egatio of there exists oe for which X is true is X is false for all. Now let us egate the sequece of statemets )-5). There is a ɛ > for which ) is false. Thus, for that ɛ ad every choice of N >, 3) is false. Choose a N arbitrarily ad cotiue. For that ɛ ad the chose N, there is a > N such that 4) is false. For that ɛ, the chose N, ad that, there is a a [, ] such that 5) is false.

32 3 MATH 57A Thus, for that ɛ ad every choice of N, ad that ad that a, we have f (a) f(a) ɛ. Let us show that our sequece f does ot coverge uiformly. Take ɛ =. (The argumet ca be adapted to your favorite ɛ.) The maximum value of f is attaied at a = /( + ), ad equals ( ) ( + ( ) ) (+) + max(f ) = =, + which teds to with, because the factor ext to the teds to exp( ), by (A-). From some o, say startig with M, this maximum value is >. The, give a arbitrary N, pick > N so large (larger tha M) that max(f ) >. Now recall that is bigger tha. Ergo, max(f ) > ɛ ad covergece is ot uiform. Note that this argumet simply says, i a roudabout way, that for every large, the graph of f goes up higher tha, somewhere. So it will ever fit ito a width strip 6. This ext propositio is oe of our goals 7. Propositio A.. Suppose f f uiformly o the fiite iterval [α, β]. The β α f (a) da β α f(a) da. Proof. Let ɛ > be give. We have β β β f (a) da f(a) da = β (f (a) f(a)) da f (a) f(a) da. α α By uiform covergece, there is a N > such that f (a) f(a) < ɛ, a [α, β]. β α So as desired. β α α f (a) f(a) da < β α ɛ β α α da = ɛ, Exercise A.. The proof used the iequality g g. Why is this true? 6 For every really really large, the graph goes up higher tha 3, ad will ot fit ito a ɛ = 3 -strip. That is aother possible ɛ 7 If you kow that the itegral may ot be defied for some wild fuctios I am ot worryig about this. Our fuctios are ice.

33 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 33 Remark A.. The itegratio iterval has to be fiite for this particular proof to make sese, sice you eed (β α) to be a fiite umber. I fact, the propositio is o loger true o a ifiite itegratio iterval. Let {, for a, f (a) =, otherwise. The f uiformly o all of R, while f (a) da = for all =,, 3,.... There is oe fial uiform covergece issue. Earlier, we had c c ad cocluded that (F cotiuous) F (c ) F (c). Now, if f f uiformly, does F f F f uiformly (F f is the compositio, F (f(a)))? It is really tedious to write dow the full statemet, so let me just idicate the idea. Suppose the Mea Value Theorem applies to F, ad that F (c) < some K o a big iterval. Thus, F (y) F (x) = F (c)(y x), or F (y) F (x) K y x for some c [x, y]. It follows that F (f (a)) F (f(a)) K f (a) f(a), ad if is large eough, the right side will be as small as you please. Of course, oe should worry about rages ad domais; the argumet that goes ito F is the value that comes out of f or f. The details are left as exercise. Propositio A.. Suppose the fuctios f are cotiuous o [ A, A] (or [α, β], more geerally) ad that they coverge uiformly to the cotiuous fuctio f o that iterval 8. Suppose the rage of the f ad f lies i some ope subiterval of [ C, C]. Let F : [ C, C] R be cotiuously differetiable, ad suppose there is a umber K such that F (c) < K for c [ C, C]. The F f F f uiformly o [ A, A]. 8 it turs out that such a limitig fuctio f is automatically cotiuous

34 34 MATH 57A A.4. Compoud iterest proofs. Lemma A.. Fix A >. Let {b k } be a sequece of fuctios covergig uiformly to zero o [ A, A]. The for every ɛ >, there is a iteger N > (depedig o A, {b k }, ɛ) such that > N implies e a ( + a + b (a) ) < ɛ, for all a [ A, A]. Istead of provig this Lemma all at oce, I will start with the simplest case, ad repeat the argumet with small modificatios. This is a bit tedious, but coceivably the strategy ca be helpful to you if you have t spet a lot of time with proofs. Proof. Let us first prove Lemma A.. Assertio (A-) will follow from the relatio obtaied by takig logarithms, (A-) lim l( + a ) = a. Recall the Taylor series for l( + x), valid for x < : l( + x) = x + x + x We use this i the left side of (A-3). l( + a ) = ( a + a + a3 3 + ) 3 ρ { }} { a (A-3) = a + ( + a 3 + a 4 + ). We eed to show that the error term ρ ca be made arbitrarily small by takig large eough. (Cautio: the is ot the summatio idex! You have a differet series for each ). Estimate: (A-4) + a 3 + a a 3 + a a + ( a ) + ( a )3 + (triagle iequality) (decrease deomiator). This last series is the familiar geometric series, r k, with r = a i our case. It coverges whe < r <, ad this will happe as soo as <. The the sum is /( r). a

35 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 35 Now let ɛ > be give. Choose N (it will deped o a ad ɛ) so that > N implies a <. The the geometric series coverges, ad moreover <. a The error term is the estimated by ρ < a a < a. Next, choose N N so that > N implies For > N, therefore, ρ < ɛ, or This is the traslatio of (A-5) a < ɛ. l(e a ) l( + a ) < ɛ. lim l( + a ) = l(e a ) ito official mathematics. The desired (A-) ow follows by takig expoetials. Next case. If you ca follow the rest of the proof, you wi a prize. I have likely made it uglier tha ecessary. So far, a has bee a sigle fixed umber. If, however, it is allowed to rage over a iterval [ A, A], we are askig about the covergece of f (a) = l( + a ) to f(a) = a. The covergece is uiform. To see this, we replace the upper bouds i the proof just cocluded by their largest possible value. Take N (it will deped o ɛ ad ow o A) so large that for > N, < A (the same as A < ). The a A <, for a A. Fially, give ɛ >, choose N N so that for > N A < ɛ.

36 36 MATH 57A The the error term from (A-3), which is ow writte ρ (a) sice a is o loger fixed, satisfies Thus we have ρ (a) < a a A A < A < ɛ. for all a [ A, A], a l( + a ) < ɛ whe > N, which is uiform covergece. Applyig Propositio A., we coclude that ( + a ) e a, uiformly o [ A, A]. Last case: the geeral statemet. Istead of (A-3), we have ρ (a) { }} { (A-6) l( + a + b (a) a ) = a + b (a) + ( + a 3 + a 4 + ), where a is used as abbreviatio for a + b (a) i the sum. Choose < ɛ <. (If we ca make somethig less tha ay such ɛ, we ca obviously make it less tha big ɛ s). Because b uiformly, there is a N such that > N implies b (a) < ɛ < for all a [ A, A] (the reaso for takig / of ɛ becomes clear shortly). For such, we have a = a + b (a) a + b (a) < a + A +. Next, take N N so that simultaeously < A+ ad (A + ) < ɛ whe > N. Just as before, Oe arrives at where ρ (a) < ɛ. l( + a + b (a) ) = a + b (a) + ρ (a), ρ (a) < ɛ, for all a [ A, A] ad > N.

37 Therefore SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 37 a l( + a + b (a) ) < ρ (a) + b (a) < ɛ + ɛ = ɛ. Aother applicatio of Propositio A. proves Lemma A. i full glory. A.5. The corollary. It is ow time to prove Corollary A.. Recall, from Calculus, Taylor s formula with remaider (also called Lagrage s formula, or Lagrage s remaider form): (A-7) f(u) = f() + f ()u +! f ()u + 3! f ()u 3 + 4! f (iv) (c)u 4 ; c is some umber kow to lie betwee ad u. Applied to cos, this gives Set u = (A-8) t : cos u = u + 4 (cos c) u4. cos( t ) = t + t4 (cos c) 4. Here, c will lie betwee ad t, ad will deped o t ad. Writig the right side of (A-8) i the form + t / + (/4)(cos c)(t 4 /), we see that we ca idetify t / with a ad (/4)(cos c)(t 4 /) with b (a) i Lemma A.. Sice t β ad cos c, it is also true that (/4)(cos c)(t 4 /) β 4 4, from which it is clear that b uiformly o [ β, β]. Thus, Lemma A. applies, ad we have proved the corollary. Primarily, we wated the limitig relatio (A-6) for the itegrals. This immediately follows from Propositio A.. B. appedix: asymptotic equivalece Here is a example. Let r (, ) ad p > be fixed. We kow that as, both r ad p ted to zero. Do they approach zero at the same rate? Oe way to see that the aswer is o is to look at the ratio r p = p r.

38 38 MATH 57A This has limit zero 9. O the other had, p ad ( + + 5) p ted to zero at the same rate, sice ( ) p p ( + + 5) = = ( + p + 5 )p, which has limit. This leads to a defiitio. Defiitio B.. If {a }, {b } are two sequeces for which a (B-) lim =, b we say that the sequeces are asymptotically equivalet, ad write (B-) a b (as, usually left usaid). I the examples above, the two sequeces both had limit zero. They might just as well both ted to ifiity, i which case the statemet (B-) would mea that they grow at the same rate. The same otatio is applied to fuctios: if f(x), g(x) or f(x), g(x) as x A (A could be fiite or ± ) ad f(x) (B-3) lim x A g(x) =, we write f g as x A. Here, we usually isert as x A, sice x might be asked to approach ay oe of may possible values A. I the case of sequeces, the subscript has to go to (uless ), so it is hardly ever ecessary to metio the fate of. Remark B.. If a ad b both have the same fiite, ozero limit L as, relatio (B-) is automatic ad cotais o iformatio. What might be useful is a compariso of the rate of approach to the limit, i.e. cosideratio of the ratio a L b L. 9 The logarithm of the right side is p l + l r. Recall that icreases much faster tha l, so the l r domiates ad it teds to. So the log of the right side teds to, ad the right side teds to zero. Calculus studets lear that geometric decay beats power growth.

39 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 39 Remark B.. The cocept of asymptotic equivalece, as defied above, is oly the most restrictive of a umber of related (ad importat) otios. Oe might woder whe the ratios i (B-) or (B-3) remai bouded i that case, the sequeces or fuctios are comparable i a weaker sese. This is writte: f = O(g) (read: f is big Oh of g ). Or oe might wat the ratio to approach zero, sigifyig that the umerator decays more rapidly tha the deomiator. Writte: f = o(g) ( f is little oh of g ). For example, the assertio e x = + x + o(x) as x meas that the error term i the approximatio of e x by + x teds to zero faster tha x, for small x. We will ot eed these variats, but they are costatly used i applied aalysis. C. appedix: Stirlig s formula C.. The formula. It is difficult to get a hadle o the growth of! as icreases. For example, 9!.48 38,! (my computer calculated 9 l k, ad the expoetiated). Is there a patter? I probability theory oe asks about the biomial coefficiet ( ) (C-) = ()! (!), which couts the umber of outcomes of the experimet of tossig a coi times i which exactly heads (ad, of course, also exactly tails) occur. How big is (C-)? Oe must replace the factorials by somethig maageable, ad Stirlig s formula provides the tool (proof below). Propositio C.. (C-)! π e. It is importat to uderstad what (C-) does claim, ad what it does ot claim. Accordig to our defiitio of,! lim =. π e It is ot true that the differece! π e is at all small. What is small is the relative error, sice! π e π e lim = lim = =.!! ( ) is also the coefficiet of x i the Taylor series of / x

40 4 MATH 57A A table will illustrate this. 5 5! 3, 68, Stirlig 8. 3, 598, Ratio Rel. Error Differece.98 3, Remark C.. Stirlig s formula (C-) ca be improved. It is possible to show that! π e [ O( 3 )] usig the big Oh otatio from Remark B.. Thus O( 3 ) stads for umbers T satisfyig T / 3 = 3 T < C,, for some fixed (but perhaps ukow) umber C. The T s are therefore smallish, but otice that they are multiplied by the huge prefactor outside the [... ]. Big Oh ca be further replaced by 39 + O( 4 ), etc. ad auseum. Math 583 deals with techiques for obtaiig asymptotic expasios like this. Example C.. To illustrate Stirlig s formula, let us fid the asymptotic form of the biomial coefficiet (C-). ()! π() () (C-3) (!) e ( =. π e ) π This applies i probability theory as follows. There are altogether outcomes of the experimet of tossig a coi times, i.e., differet sequeces of H ad T (heads or tails). If the coi is fair, each sequece has probability /. Therefore, the probability of gettig exactly heads is ( ). π E.g., the probability of gettig exactly 5 heads i tosses is about / π.38. Not likely. So the questio arises: what is the law of averages, precisely? Oe versio of it, illustratig highdimesioal geometry, is preseted i Sectio 5, see Remark 5.. Exercise C.. Use Lemma 3. to derive the asymptotic approximatio i (C-3), thereby circumvetig Stirlig s formula.

41 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 4 C.. Proof of Stirlig s formula. This subsectio is tacked o oly because a proof of this useful formula is rarely ecoutered except, perhaps, i a more advaced course that icludes asymptotic methods. Oe stadard approach is via the Γ-fuctio, Γ(x) def = t x e t dt. Oe shows, usig itegratio by parts, that Γ( + ) =!, for =,,,.... O the other had, there are techiques for approximatig itegrals like this oe for large values of x, ad those yield Stirlig s formula (as well as the improvemets i Remark C.). There is, however, a elemetary proof as well (elemetary easy), which relies o somethig called the Euler summatio formula ad Wallis product (3.9). I follow K. Kopp, Theory ad Applicatio of Ifiite Series, 64. We wat to prove that lim! =, π e or the equivalet statemet for the logarithms, (C-4) lim ( l k l π ( + ) l + ) =. The sum k= l k k= is roughly comparable to l x dx = (x l x x) = l +. We begi to see where Stirlig might come from, but evidetly the approximatio of sum by itegral is too crude. That is where Euler s more refied method comes i. We eed the greatest iteger fuctio, defied by [x] = greatest iteger x. Propositio C. (Euler summatio formula). Let f be cotiuously differetiable o [, ). Abbreviate f(k) = f k. The for =,,..., (C-5) f + f + + f = 57 Notes, pp f(x) dx + (f + f ) + (x [x] )f (x) dx.

42 4 MATH 57A Notice that the first two terms o the right already give (for f = l) ( l + ) + ( + l ) = ( + ) l +. Proof. For k =,,...,, we have by itegratio by parts: (C-6) k+ k (x k )f (x) dx = [(x k )f(x)]k+ k = (f k + f k+ ) k+ k k+ k f(x) dx. f(x) dx Sice k x k + i the left side of (C-6), we may replace the k i the itegrad by [x] (the fact that [x] k at the upper limit, where x = k + = [x], does ot affect the value of the itegral). Thus (f k+f k+ ) = k+ k f(x) dx+ k+ k (x [x] )f (x) dx, k =,...,. Summig these equatios over k, we get o the left side (f +f )+ (f +f 3 )+ + (f +f ) = f +f +f 3 + +f + f, ad o the right side f(x) dx + (x [x] )f (x) dx. Add (f + f ) to both sides to get Euler s formula. Notice that P (x) def = x [x] is a periodic fuctio, period, with jump discotiuities at the itegers (Figure 6). We are about to itegrate by part i the itegral P f i Euler s formula; i preparatio observe that P is the derivative of the periodic bouded fuctio (Figure 7) P (x) def = (x [x]) (x [x]) +. P has corers at the itegers, as expected, sice P has jumps there. Note also that P (k) = / for k =,,,.... Lemma C.. If f exists, the P (x)f (x) dx = (P (x)f (x)) + P (x)f (x) dx.

43 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 43 x-[x]-/ Figure 6. The fuctio P i Euler s formula..5*(x-[x])^-.5*(x-[x])+/ Figure 7. Atiderivative of P Let us ow set f(x) = l x. We already have gotte to (C-7) l k = ( + ) l + γ, k=

44 44 MATH 57A where, by the lemma, γ = P (x) x + P (x) x The itegral coverges as, by compariso with x dx. The term before the itegral is (C-8) P () P () =,. We coclude that γ has a limit γ as. The last step is to show that γ = l π. The method strikes me as beig exceedigly tricky. You have to kow what you are lookig for, or be very clever. Go back to (3.9), or (C-9) lim p lim p dx. 4 (p) 3 5 (p ) p + = π 4 6 (p) 3 5 (p ) We are about to work γ ito this. By (C-7), π = p +. l( 4 (p)) = l( p p!) = p l + l p! Also, = p l + (p + ) l p p + γ p } {{ } from (C-7) = (p + ) l p p l + γ p. l(p + )! = (p + 3 ) l(p + ) (p + ) + γ p+. Subtract. O the left side, you get a differece of logs, which is the log of the quotiet 4 (p) 3 4 (p + ) = 4 6 (p) 3 5 (p ) p +. The expressio o the right is rearraged a bit (check it), resultig i ( ) ( 4 6 (p) l = (p + ) l ) 3 5 (p ) p + p + l(p + ) + l + γ p γ p+.

45 SOME GEOMETRY IN HIGH-DIMENSIONAL SPACES 45 Move l(p + ) to the left side (where it produces a p+ iside the log). Now everythig i sight has a limit. The left side teds to l π by (C-9), the term ( (p + ) l ) p + teds to by the compud iterest formula (A-9), ad γ p γ p+ γ γ = γ. (Here it is importat to kow that the sequece {γ } has a limit that was the reaso for brigig i P (x)). Thus π l = + l + γ, from which (!) γ = l π. Remark C.. There is a improved Euler formula. Namely, oe ca cotiue with the itegratio by parts: P f = [P f ] P f = f [P 3 f ] + = f [P 3 f ] + P 3 f P 3 f ; here P 3 (x) = 6 (x [x])3 4 (x [x]) + (x [x]) is the bouded periodic atiderivative of P. I this way, oe gets the term, ad subsequet terms, i the improved Stirlig formula metioed i Remark C.. You ca see the appearig i (C-8).