ON MILMAN'S INEQUALITY AND RANDOM SUBSPACES. Y. Gordon* Technion - Israel Institute of Technology Haifa, Israel. Abstract

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1 ON MILMAN'S INEQUALITY AND RANDOM SUBSPACES WHICH ESCAPE THROUGH A MESH IN ~'* Y. Gordon* Technion - Israel Institute of Technology Haifa, Israel Abstract Let S be a subset in the Euclidean space/r" and < < n. We find sufficient conditions which guarantee the existence and even with probability close to, of - codimensional subspaces which miss S. As a consequence we derive a sharp form of Milman's inequality and discuss some applications to Banach spaces. Introduction Consider the following theorem which we label as the "escape" phenomenon of random -codimensional subspaces through a mesh S. Theorem. Let S be a subset of the unit sphere S~ - of R n and associate to S the number -- a, f sup(x,u)m,~_l(du), where m,~-i is the normalized rotation invariant measure on S~-I zes the sphere and If s < a ( <_ < n) then there is a -codimensional subspace which does not intersect S. This theorem and other results developed here, are based on two theorems A and B on Gaussian processes, which were originally proved in IG] and generalizations proved in [G3]. Using Theorem A, we obtain in section, as a simple corollary to the "escape" phenomenon, a sharp new proof of an inequality of V. Milman and its refinements due to Pajor and Tomcza. The inequality, stated here in Theorem.5, has some important applications to Banach space theory. One of them is the theorem due to Milman IM, M2]: * Supported in part by the USA-Israel Binational Science Foundation (BSF) grant ~ Supported in part by the Fund for the Promotion of Research at the Technion ~ and the V.P.R. grant #

2 85 Theorem. For any 0 < A <, any Banach space X of sufficiently large dimension n has a subspace of a quotient of X of dimension >_ An, which is f(a) isomorphic to a Hilbert space. As we shall see in section, this and other results can easily be derived from the inequality. In section 2 we combine the escape phenomenon and L~vy's isoperimetric inequality to prove the following sharper form of a fundamental result stated in section 2 of [MS] and originally proved in [M5]. Theorem. Given a function f(x) E C(S -) let M/ and wf(e) denote the median and mod- ulus of continuity off. If O < e <_ and = [ne2 / 4] - >_ O, then there is a subspace E of rt-- dimension such that If(x) - M/t <_ wf(2 ) for every z E E fq S 2. This eliminates the log term in the estimate g = [n 2/(2 log 4/ )] of Theorem 2.4 of [MS]. In order to obtain measure estimates as well, it is necessary to replace Theorem A by its quantitative analogue Theorem B, and combine it, with L6vy's theorem or Pisier's theorem, or as we shall do here, with Theorem 3.2 due to Maurey on the tail distribution of a real Lipschitz function defined on the Gaussian probability space (frith,p). Thus we are able in section 3 to estimate the measure P of the subset of all -codimensional subspaces of the Grassman manifold ~n,n- which miss the set S + K, S being an arbitrary closed subset of S~ - and K a closed convex subset in ~7~ '~ which contains the origin in its relative interior. A special case is K = rb~, the Euclidean ball of radius r. P is then the probability that a random -codimensional subspace will be at Euclidean distance greater than r from S; if r = 0 this simply means that ~ will miss S. We shall see for example that if r = 0, P is very close to if s < v~- c for large and suitable large constant c. In section 4 we consider a finite sequence of N arbitrary convex sets N { Si}i=l situated at various locations in ~r~. We are interested in estimating the probability that a randomly chosen -codimensional subspace will for all i =,...,N pass at a distance greater than di (> 0) from Si. This probability can be estimated by an expression which involves only the numbers di, ti, Ti and si (i =,..., N), where ti and Ti are the smallest and largest distance of the points of Si to the origin and si is the number s we associate above with the set S = Si. We shall see in section 5 that some of these results admit generalizations. In order to do that we shall also need to extend Theorems A and B. For example, we consider a finite sequence of L arbitrary subsets of the unit sphere S~- and we find conditions which imply that there exists a subspace of codimension which misses at least one of these sets and estimate the probability of this event.

3 86. The Escape Phenomenon and Milman's Inequality Theorem A below is the ey to the existence theorems contained in sections and 2 and was originally proved in [G]. We shall see in section 5 that Theorem A is a particular case of Theorem C, which was proved in [G3] by a different and simpler method. Theorem A. Let {Xiy} and i =,2,,..., n, j =, 2,,..., rn, be two centered Gaussian processes which satisfy the following inequalities for all choices of indices: (i) Elxij - Xil ~ <_ IEIY<~ - Y.,I 2, (ii) EIX O. - Xel 2 >_ EIYi j - Y~l 2, if i ~. Then, E ~n %ax Xij <_ E min max Yij. i ] Taing n = in Theorem A we obviously obtain the Fernique, Sudaov inequality [F,F2]. Corollary.. Let {Xi} and {Yj}, < j < rn, be two centered Gaussian processes such that F, IX j - Xl 2 < EtY j - Yl 2 for all j,. Then,/EmaxXy _</EmaxYj. 3 2 Throughout, we shall denote by {gii}, {hi}, {gj}, {g} independent sets of orthonormal Gaussian r.v.'s and G = G(w) = go(w)ei e j i: ]= will denote the random Gaussian operator from ~, to /R, where {e,}i=l is the usual unit vector basis of ~", and II " I]2 stands for the Euclidean norm on ~'~. The letter c will denote constants. Corollary.2. Let S C J~:C ~ be a closed subset, < < n. Then (.) (.2),,-,,,,/2 + Eh,x,}) E( min fig(w)(x2) > ~:(min { II~ll~ ~ Z.., g~j -- " xes E(m~lla(o.,)Cx)ll2) xes -- x6s > a min II xl2 - IE( max ~ h xi) -- xes xes ~ ' < E(max {ll:':ll~t 2_..., gj),~"~ 2,/2 + Ehizi}) n -- xes z6s where.~ = E(Eg]) ~/~ = v~r (~) /r (}) satis[ies /v/+ < a < v/-.

4 87 Proofi For x C S and y C S - - {y C ~ ; IMI= = }, we define the two Gaussian processes and x~,~, = Ilxll~ ~ gsyj j:l i: h,~, i= 2= It follows then that for all x,x' 6 S and y,y' E S2 -, we have ~_,IX~,~ - X~,,y,[ ~ - EIY,,y - y~,,y,i 2 = I:~~ + I:~'~ - 2~ll=II~:'ll=<y,y'> - 2<x,x~'>( - <y,y'}) x t > lt~tl~ + II~'ll~ - 2il=ll=ll:~'ll=<y,y'> - 2I~= I=( - <y,y'>) >0. with equality if x = x'. By Theorem A this implies that and by Corollary. (min max Xx,~) < E(min max Yx,~,) ~s yes~-* -- ~cs yes~ -~ /E(maxXx ~) > /E(maxYx,y). xjy ~ -- xd/ The last two inequalities imply inequalities (.) and (.2). The estimate for a follows from a~ _< /E(~ gy) = and av'- + > aa+, =. n Remar (.). Using the identity aa+l =, it follows by induction that ( ) a = V~ -- -~ + 3-~ + O(-s) " Corollary.3. Let S be a closed subset of the unit sphere s~ -* = {:~ ~'~ ; I== = } and s = E(max~hixi). zes If < < n satisfies a :> s, then there is an operator T : ~ --+ ~ such that (.3) IITxIJ2 < a + 2~ for all x,y C S. IiTYII~ - a - s Proof: By Corollary.2,/E(minxes llg(w)(x)ll=) _> a-s > 0and E(maxxes llg(w)(x)h2) _< a + s. Hence, there exists w0 such that T --- G(wo) satisfies (.3).

5 88 n-- There is a natural upper bound for s which is obtained by taing S = S 2 Integrating over the unit sphere we get that s can be expressed by s = an f r*--i where rnn_t is the normalized rotation invariant measure defined on S~., thus s < an. m~s (x, u}rnn_l (du) Another estimate for s can be obtained, for example, by using Dudley's theorem [D]: Let N(e) be the number of elements in a set S C ~'~ which forms an e net in the II II 2 norm for oo od S. Then, s < c y x/log N(e)de. Thus, s can be replaced by c f lobe)d6 in Corollary The escape phenomenon mentioned in the introduction is summarized in the next theorem. Theorem.4. Let S c IR n be a compact subset, < < n and G(w) : ~ be the Gaussian map from R n to R. i=j= ~ gij(w)ei ej () F, m~ [[G(w)(x)[[2 > 0 iff there exists a subspace of codimension which misses S. r~ (2) IfS C S~ - and a > s = E(max~l, then EminllG(w)(x)ll2zes > o. Proof: The subspace E~ = G(w)-i (0) has codimension a.e., hence if/e xn~ IIG(~)(x)l2 > 0 then there is an Ewo of codimension which misses S. Conversely, every -codimensional subspace has the form E~ and if E~ o misses S, then compactness of S implies that E~ misses S, i.e., rain IIG(~)(x)ll2 > 0, on a set of positive measure; this proves (). Statement (2) follows xes from inequality (.) a Given a closed convex body K in/r n which contains the origin in its interior, we associate with K the dual body K* = {x E /I~ n ; (x,y) < for all y E K} and the "norm" IIxl[g = n inf{t > 0 ; x E tg}. E(K) will denote ~7 ~g, eillk and 2(K) = max{llx[[2 ; x g*}. If g is centrally symmetric, then II IlK is a norm and the normed space X -- (~", II * IlK) has K for its unit ball; we shall then refer to E:(K) by/e(x) and e2(x) = e2(k). Theorem.5. Let K be a closed convex body in ~, 0 6int K, < <<_n and let Kr = r- l K N B~ (r > 0). Then there is a subspace E of codimension such that ~2 _< r0~llk for all x E, where r0 = inf{r > 0 ; a >/E(K;)} < a~llf~(k*). Proof: Let S = S~ - f7 Kr. By Theorem.4, if a > s = le(sup~-~xih~) then there xes is a -codimensional subspace E which misses S, hence if x E n S~ - then x (~ r-lk, that is rll~llk _> IlxlI~ for all x E. Notice that s < /E( sup Exihi) = /E(K;). To xek~ prove the inequality for ro, since Kr C r-lk, K; D rk*, therefore rllxllk- --< ]IxlIK" and E(K~) <<_r-~ie(k*), which implies r0 _< alj~'(/'(*). El

6 89 Remar (.2). Originally V. Milman was the first to prove that there is a function : (0,) ~ ~+ with the property that in every Banach space X of dimension n there is a subspace E of codimension < ~n such that IIxt[2 _< ( )n-/2e(x*)iixll for all x E E and the estimate (e) < c - was proved in [M]. Pajor and Tomcza [PT2] improved the estimate to ( ) < C -/2 where C is a universal constant. Their original proof is based on L~vy's isoperimetric inequality [MS] and Sudaov's minoration theorem IF2], an earlier version of which appears in [PT}. It is worthwhile to recall here some consequences of Theorem.5. We first give the operator formulation of the theorem. Let u : X --+ Y be an operator between the Banach spaces X and Y. The th Gelfand num- ber of u is defined by o(u) = inf{[]u [ Eli ; E C X, codime < }, =,2... For u: ~ --* Y, e(u) : Ell (usually the norm of u, (u) is defined by (~ /2, but by Kahane's inequality [MS] the two expressions are equivalent). For u : ~2 --* Y, ~(u) is defined as sup { (uv) ; [[v : l~ -* 2[ - }. Theorem.5. can be reformulated as follows: Theorem.5'. Let u : X --~ ~ and X, = (X, [I * lit) be the space normed by t[z[[, = max{rhx H, Iluxll2 } (r > o). Let < < n, then C.i.l(U) ~ r'0 = inf {r > 0 ; (u* : /~ ---r X;) < a}. In particular, ac+l(u) <_ e(u* : e~ ~ X*). Proof: Without loss of generality we can assume that u is invertible and define on/r '~ the norm Ix[ = liu-~xll, then u: g ~ (~n, I " I) is an isometry. Now let Ixlr = max{fix I, IIx2} and apply Theorem.5 to the unit ball Kr of the normed space (/R '~, I* [~), with K equal to the unit ball of (~'~, I * I)- We thus obtain a -codimensional subspaee E for which ~2 <- ~0~ for all x E E, where F'o = inf{r > 0; a >_ ~7(K;)} = inf{r > 0 ; t(u*: ~ ~ X~ < a}. o Theorem.6. Let X = (~iv', t * II) be an n-dimensional Banach space and T2(X*) and C2(X) be the Gaussian type 2 and cotype 2 constants of X* and X respectively. every A E (0, ), there exists a subspace E of X of dimension [An] such that (i) d(e,e~ ~") <_ ( - :~ - n-~)-~/2t~(x'), if n -~ < ~ < -- n-i; 0i) d(e, e~ ~n]) < c( - ~)-'/2C2(X)log ( + c( - A)-'/2:C2(X)), ifn >_ N;~. Then for Proof: (i) without loss of generality assume B~, the unit ball of ~, is the ellipsoid of maximal volume contained in Bx, the unit ball of X, and let + -- n - [An]. By Theorem.5 there exists a subspace E of codimension + such that [Ixll _< Ilx[2 < a;:,e(x*)llxll for all x e E.

7 90 Now, a+l _> v~ :> x/~( - ~ - n-') /2 and by Lemma 4. of [BG], /E(X*) _< v/-nt2(x*). This proves (i). (ii) Since T2(X*) <:_ C:(X)K(X) < cc2(x)log (d(x,g'~) + ) where K(X) is the K convexity constant of X (cf. [P] or [MS] and Remar 2.2 [P2]), the required result follows by applying the iteration procedure of Milman [M3] a finite number of times (with an appropriate sequence of Ai's in (0,)) to a subspace of a subspace..., etc., of X. o Remar (.3). The fact that (ii) has linear dependence on the cotype 2 constant C2(X), except for a log term, was first proved in [M, M3. In the form stated above, (ii) was established in [PT2] by means of the sharp estimate ( ) < ce -~/2 obtained there. The following theorem is due to Milman and appears in the form stated here in [M4]. Theorem.7. Let f(a) = C(--~)-- log (C2/(--.~)), A E [,). Then, given any~ e (:,), every Banach space X of dimension n(> N~) contains a subspace of a quotient of X, E, of dimension [An], such that d(e, ~]) Proof: _< f(a). Given any ~ <: _ AI <:, we apply Theorem.5 to find a subspaee El of X* of dimension > [Aln] such that "x"2 < v~n/ev~l,,x,, for all x E El. Dualizing this inequality and applying Theorem.5 again, we find a subspace E2 C E[ of dimension :> [A~n] such that both inequalities imply HxH2-< v/'~---'((e-)'~zln--z n ']xh for all x ee2, d(e2,~ime~ ) < ]E(E)~:(X) <: 2/E(X)]/~(X*) By a result due to Lewis ILl, there exists u: l~ --~ Z for which ~(u) *(u -~) = n, and by [FT] (u *-) < c~.*(u-')g(z) and now applying the inequality K(X) < clog (d(x, ~) + ), we obtain by choosing the proper ellipsoid in ~, namely the Lewis ellipsoid d(e2,e~me~) < clog (d(x,~'~)+ ) - l - A ' where E2 is a subspace of a quotient of X of dimension ~ A~n. Replacing X above by E 2 and continuing in this manner, we obtain the result after a finite number of iterations and the proper choice of the sequence A~. In order to obtain a subspace of a quotient of X which is +~ isomorphic to a Hilbert ---'7 space for a given ~ E (0, ), we shall use the following theorem proved in [G], see also [G2]. D

8 9 Theorem.8. Let Y = (/R ", tl. ti) be a Banach space and 62(Y) = m~o(tiytl/ltyit2). < m < n satisfies F,(Y) > V/-m~2(Y), then Y contains an m-dimensional subspace F with ~(Y) + V~a~(Y) /E(Y) - ~'~ 2(Y) " If Theorem.8 combined with Theorem.7 provides immediately and obviously the +~ version of Theorem Let 0 < e <. There exists N such that for any n(>_ N) dimensional Banach space X, +~ isomorphic to a there exists a subspace of a quotient of X of dimension > cc2n which is i-:-g Hilbert space. Proof: Tae A = /2 in Theorem.7 and apply Theorem.8 to the space E of dimen- sion In/2], to find a subspace F c E of dimension m which will be ~ isomorphic to g~' provided x/~e2(e) <: /E(E). But since d(e, ~ '~/2]) <_ f(), it follows that ~2(E)/E(E) < f()/e(~'~/2]) ~ cn -/2 and this yields the estimate for m. [] Remar (.4). In a similar manner we can obtain ~ versions of Theorem.6. l+e 2. Applications to Continuous Functions on the Sphere Given a set A C S~ - and ~ > 0 we denote by A~ the set {x e S~ - ; p(x,a) <_ ~}, where n- The median Mf of a function f E C(S~ -) is a number such p is the geodesic metric on S 2. that m,-l(f <_ My) > ½ and mn-(f ~ My) >_ ½, where rnn_l is the normalized rotation invariant measure of S~ -. L~vy's classical isoperimetric inequality states (cf. [MS]): (2..) Let f e C(S~ -~) and A = {x C S; - ; f(x) = My}. Then m,~_l(a,) >_ - V/-~exp ( - ½(n - )e2). Let wf(e) -- sup{if(x) - f(y)[ ; p(x,y) _< e} be the modulus of continuity of f. It is clear from the definition of A in (2.) that [f(x) - My[ _< wy(~) for every x E A~. Thus by L6vy's isoperimetric inequality, f(x) is close to My on the set Ae which has large measure if n = is big. Theorem 2.. Let A c S~ -, 0 < 6 <_ and g = [½ne2mn-x(A)] - _> O. Then there exists a subspace E of dimension g such that E N S~ - C A,.

9 92 Proof: We apply Theorem.4 to S = (A~)c, the complement of A~. By standard integration n s = E(sup x,h, / sup(x,u)m _,(du) es-: ) = a. J g Notice that ifu E A and x C S, then (x,u) <_ cose and for all other u E S~ - (x,u) <, hence s <_ a (mn_l((a) c) + mn-l(a)cose) = an( - ( - cose)rnn_,(a)). Let = n - ~, then (2.2) a/a. > V( - )/n >_ ( - ½e2rn,~_,(a)) :/2 ~4..'~ /2 >_ (- 2e2rnn-a(A) + ~rnn-l(a)) > - 62rnn_l(A) > - ( - cose)rnn_a(a) _ s/an, therefore a > s. By Theorem.4 there exists a subspace E of dimension which misses S, hence E n S~- C A~. [] Theorem 2., combined with the isoperimetric inequality (2.), sharpens the estimate obtained in Theorem 2.4 of [MS I. Corollary 2.2. Let 0 < ~ <_ and ne 2 >_ 4. Then for any function f E C(S~-I), there exists a subspace E of dimension >_ [ne2/4]- such that (2.3) If(x)-Mfl < wi(2c) for every xeens~ -. Proof: If B = f-a(mf), then by the isoperimetric inequality, rnn_x(b,) >_ - v/~exp ( - (n - )e2/2) > ½ since by our assumption ns 2 > 4. Hence taing A = B~ the dimension l of the subspace E in Theorem 2. is > [n~2/4] - and E N S: '-I c B2,. a 3. An Isoperimetric Inequality on the Sphere of ~c~n In this section we shall mae use of Theorem B proved in [G]. Theorem B turns out to be essential for obtaining the measure estimates in this section. A simpler proof of this theorem appeared recently also in [K], and other related results, as well as generalizations in various directions, may be found in [G3].

10 93 Theorem B. Let {Xi/} and {Y/j}, < i < n, <_ j < m, be two centered Gaussian processes which satisfy the following inequalities for all choices of indices (i) E(X~) = ~:(Y,~) (ii) ~(x~iz~) > E(Y~sY, ) (iii) E(XijX, ) < E(Y~sY, ) if i # l. Then, P(NU[xq > Aol) < P(NU[Yo -> A/j]) for all choices of Aiy /R. ij ij Lemma 3.. Let G(w) = ~ gq(w)ei ej : R '~ -+ N and S C ~'~ be an arbitrary i= j=l subset. Then for all choices of real A, (x S), ~>( N [a(<4(~)2 + I~~, ~_ ~,:)~_ P( N [II~II~(Z ~J) '/=+ ~,h, ~ ~,:]). z~s z~s Proof: For x C S and y 6 Sff - we define the two Gaussian processes, Z~,~ = (G(w)(x),y) + nx[2 g and Yz,y = [[x[[2,).., "~--" giyj + ~ xi hi where hi, gi, g denote orthonormal Gaussian r.v.'s. It is easy to chec that/e(x~,u) = E(y2,u ) and E(X=,vXz,,v, ) - E(Y=,yYz, v, ) = ( X,.T' >(y,y l > + IIxlI2IIx'H2 - tlxh2ix'n~(y,y') - (x,x') = (t{~n~l;~'li2 - <~,~'>)(I - <y,v>) which is always nonnegative and equal to zero if x = x I. For each x S the set U [X,,y _> A,] = [UG(x)n2 + gllxl[2 >_ A,] is closed in the yes~- probability space {/R n+l, P} where P is the canonical Gaussian measure of j~n+l. Hence N U [xz,y _> A=] is closed. The same can be said for the corresponding expression with xes yes~- I the Yx,~. By Theorem B, for each finite set {xi}~ c S we have N P( N U Ex=,,~ >_ ~,:,) _> P( N U EY~,,~ >- ~,=,) i=l ~s~-~ i=i ~es -~ and so, ordering the collection of finite subsets of S, F, by inclusion, we obtain that the limits exist and satisfy the inequality N lime( N U Ix=,,~ >_ ~=,) >- l~e(n U Ir~,,~ >_ ~:,) i=l ~es~-* i=l ues~-' N N

11 94 But as the sets n U [x,,y >,,]) and n U [Yx,y_>,,] are closed and P is a regular z6s yes~- ~ zes yes~measure, it follows easily that the two respective limits over F are equal to and satisfy the inequality ~es yes~- ~ ~es yes~- We shall also use the following theorem due to Pisier on the normal tail distribution of a Lipschitz function [P3]. Pisier proved in [P3] a more general result with the constant = 2= -2. Theorem 3.2. Let f: R ~ ~ R be a Lipschitz function satisfying IS(z) -S(Y)t -< all x- Y]I2, let c = and P be the canonical Gaussian measure on ~n. Then for all A > 0 P(f(x) - let > A) < exp(-ca2/a 2). Remars (3.). Let ~n,n- denote the Grassman manifold of -codimensional subspaces of/r n and 7n,n- the normalized Haar measure on ~n,n-. If S C/R n is a subset, a subspace E..6n,n-- meets S if and only if it meets Srad, which will denote the symmetric radial projection of S u (-S) onto the unit sphere S~ -. When = n - and S is a measurable subset of R", qn,l(~ E ~n,; ~ N S ~ ) = mn-l(srad), where ran- is the normalized surface n-- area of the sphere S 2. Let E denote the set of all symmetric measurable subsets of S~ '-, in particular Sr~d E :E if S is a measurable subset of/r n. "7... induces the measure ~/... (~ E G... ; ~ N S ~ 62) on the elements S E E. This is a probability measure which is invariant under orthogonal transformations u E On applied to S. Similarly, if G = G(w) = ~ ~ gi,j(w)ei e i is the Gaussian operator from/r n to/r, /=lj=l then a.e. G(w)-'(0) is a -codimensional subspace of ~n and P(w : G(w)-l(0) N S ~ (I)) is also a probability measure on S E ~, which is invariant under orthogonal transformations u E On; because, Gu and G have the same distribution for every u E 0,~. By uniqueness of the Haar measure, the two measures are identical on E, i.e., ~/n,n-}(~ E ~n,n-- ; ~ N S ~ (I)) = P(w ; G(w)-t(0) N S # (I') for all S e E, hence also for all measurable subsets S in/r". (3.2) Let K c/r r' be a closed convex subset which contains the origin in its interior. Let I]GII be the "norm" associated with G as a map from (/R", I] * IlK) to e~ = (R, [] * ]]2). By Corollary.2(2), taing S = K, we have and conversely, if x E K, then ~llall < a~(g*) + ~(K*)

12 95 implies EIIGII >_.EtlG(x)lh = (~-]~,) ~Z..,gJ; n /EIIGII _> ae2(k*) ; moreover, Tt that is, max {ae2(k*),le(k*)} <_ m:llcll _< ~~:(g*)+/e(k*). The following theorem estimates from below the measure of the set of -codimensional subspaces ~ C 9~,,~- which miss a given closed subset S in o 2~''~- by a distance, with respect to the II" IlK "norm", greater than : Theorem 3.3. Let S C S~ - be a closed subset, K c ~ n a closed convex set with OeintK and s = ~(max~z,h,). x6s Xra( - ~(K*)) - ECK*) > s, < < n, then ([ ]) 7 a( -- e2(k')) - J~(K') - s Proof: Let T = ~ tijei ej and litll be its norm as a map from (/R",K) to /~. It is 4:I j:l easily seen that I]ITI}- lit'hi < e2(k*)(~(t 62(K*), so by Theorem 3.2 t,2 O - t~j) 2) /2 hence HTI] has Lipschitz constant P(IIGII ~ ( + e)/eiigii ) < exp ( - ½(e~IIGII/~2(K*)) 2) ~ e(-e2a2/2), where E > 0 is to be chosen later. Let A = ( + E) IIGII and P = P(w ; G(w)-l(0) CI (S + K) = 0). Setting Q = P(w ; minllg(x)ll2 ~ )0, we have zes Q <_ P(w ; mhn IIx- ylikhgll > A) zeg- ~ (O),yES rain x ) < P(tlall > ( + ~)~:IIClI) + P(, tl - YlIK > zeg- (O),y6S = P(ttvll _> 0 + O~ttcH) + P -< exp(-~au 2) + P"

13 96 On the other hand, Q+~I exp(_v2a~/2). > Q +. PEg > ~a). >. P( ~ [ItC(~)ll~ + g > ~ + ea]) zes and by Lemma 3. n --> P( A [(~'~g. 72)/2 -~- E xihi>- ~-~a]).~r. zes I Moreover, since the Lipschitz constant of the function (~ g2)*/2 is, it follows by Theorem 3.2 that -R=P(U rt [(~g~)l/2+~_xihi<a+ca]) xes < P((~j~g])l/2 < (- ~)a) + P( U [ xihi <_ A-- (-- 2 )a]) :~ES r~ < exp(- 2a~/2) + P(max~xihi > ( - 2e)a - A). zes "---" Now, the function f(h) = max~ xih defined for h E /R n, has Lipshitz constant equal to, xes E(f) = s and since for the proper choice of E > 0 a( -- 2~) -- A -- s > a( -- 2E) -- s -- ( + )[ae2(k*) + ~(K*)] > 0, we obtain by Theorem 3.2, P(f(h) > a( - 2E) - A) < exp ( - ½{a( -- 2e) -- A -- s} 2). Combining the inequalities we have P=~... (~E~... ~;~n(k+s)= ) ~_- ~5 exp(_a~e2 /2) _ exp(_a2/2) where o = a( - 2~) - ~- ( + ~)[a~:(k*) + ~(K*)]. We now choose ~ so that ~ = o, i.e., a( - e2(g*)) - IE(K*) - s ~a = 3 + e2(k*) + IE(K*)/a > 0, then the required estimate P > exp(-e2a~/2) follows. D

14 97 Remar (3.3). By continuity Theorem 3.3 remains true if K has dimension lower than n, i.e., when K is a closed convex set which contains the origin (not necessarily in its interiooand dimspan (g) < n. The values e2(k*) and/e(k*) are then understood as ~2(K*) : max{llxl]~ ; x E K}, and r~ ~,(K*) = E( max ~-'xigi). xek ~-~ l Taing for example K -- rb~ -- {x -- (xl,.., xt, ) ; ~ x~ _< r 2 ), we obtain ~2 (K*) -- r, IE(K*) -- rae and so, if <, _< n and r satisfies a( -- r) -- rat > s, then the measure of the -codimensional subspaces ~ E-.~... which miss the set S -t- K is greater than In particular, taing r = 0 we obtain: 2 2 L 3+r+rae/a n Corol,ary 3.4. Let S S;-' be a dosed, set, s = <_ < n and sume a > s. Then 3... /~(~E~... ; ~MS=(I))>I _ - ~ 7 exp (-l(a - s) 2 ). Corollary 3.5. Let S C S~ - be a closed set ands = E( max ~ xihi). Ill < < n, a > O xes and s < ~- 3a, then 7 exp(_o~2/2 ) -~- 0(_/2)... (~E ~... ; ~AS---- ~) ~_ - ~ In particular, if s < V~- 0(n--/2). 3a, then S u (-S) has surface area smaller then 7 exp(_a2/2) + Proof: Apply Corollary 3.4 and use the fact that the surface area of S u (-S) is ~.,(~E~.,, ~ns# ). Remar (3.4.). If 0 < A < and 0 < r < ~ are fixed and s < A( - 2r)v/-~, then taing in Theorem 3.3, : n - and K : rb~ we obtain?n,,(~ E ~,,, ~ N (S + rb~) (I)) < c exp ( - f(a,r)n), where Cl is a constant and f(a, r) a positive function. This says that the surface area m,~-i ((S4- rb~)rad) of the radial projection of (S 4- rb~') U (-S + rb~) onto S~ - tends to 0 exponentially fast with n. A more careful analysis of this situation is as follows:

15 98 r~ Corollary 3.6. Let S C S~ - be an arbitrary closed set, s = E( max ~ xihi) and t be any xes number such that s < t < ~-n. Let Kt = 7,- [ tn-/2)b~. Then, there are universal constants el,c2 > 0 such that, rrtn_x ((S -}- Kt)rad) ~ Cl exp ( - c2(z - 8)2). Proof: We apply Theorem 3.3 with = n-. Since an = x/-ff( - 4A~ +0(n-2)), an- = ~(--~-~n+0(n-2))=~/-n( -- 4-~+0(n-2)). We also have e2(k;) = ½(-tn-/2) and /E(K;) = ½( - tn-u2)an and the inequality in Theorem 3.3 yields the result. [] Combining the estimates of Corollary 3.4 and Theorem 2. we obtain a quantitative version of Theorem 2.. Corollary 3.7. Let 0 < ~ _< and f+ _< ½ns2rnn_t(A), where A C S~ "-~ is an arbitrary closed subset. Then Q'n,l(~ e ~n,~ ; ~(-I S2 - C A,) _~ - c, exp { - c2~ -l(lne2mn_l(a ) - e- ) 2 }, where cl and c2 are positive universal constants. Proof: Let = n - ~, By inequality 2.2 with S = (A~) c we have _> {- -n(- ½e'rnn_l(A))}/2v/-n >_ {½n 2rn,~_,(A) - e- }/2v/-n > 0 and Corollary 3.4 concludes the proof. [] Remar(3.5). Corollary 3.7 and L~vy's isoperimetric inequality imply that if ne 2 is big, then the set of subspaces E of dimension ~ n 2 for which (2.3) is satisfied has probability close to. 4. Missing Convex Sets by Random Subspaces Let {Ki}~ be a sequence of N closed convex sets in /R n which contain the origin 0 in their relative interiors and let {zi} N be arbitrary points in/r n. We denote by ti (resp., T~) the smallest (resp., largest) Euclidean distance of a point in zi + Ki to the origin and set /E(K*) =/E( max ~ xjhj). In the next theorem we estimate from below the measure of the xek~.4= set of all -codimensional subspaces ~ in ~n,n- which are distant di at least from zi + Ki for all i=,2,...,n.

16 99 Theorem 4.. q... {~ E ~n,- ; d(~,z, 4- K,) > di for all i :,2...,N} _> - exp(-~:a~/2) N - (3/2) e p(-~:a~/2) - Eexp( - {( - 2 )ati - E(K;) - ( 4-5)di(an 4- a)}~/2t 2), i= where ~,,~ > o a~a where we ~s~ume ( - 2~)at~ > ~,(K?) + d~( +,~)(an + a) ~o~ ali i= l,...,n. Proof: Let Ilall be the norm of a = ~ gij(w)ei e i as a map of ~ to e. We have i=.i=l an <_ EIIGI I <_ an 4- a. Let Ai = di( 4-5)/EIIG H and P = "... (~ E ~... ; d(~,zi + Ki) _> di, =,2... N) = ~(~ ; ~(a-l(o),z, + K,) >_ ~,, ~ =,2,..,N) Observe that Q:= P(Vi min IIC(x)ll~ > A,) xezi +Ki < P(Vi, IlClld(G-l(O),z~ + K~) > A,) _< P(IIGII _> (x +,~)~llall) + P(Vi, d(a-l(o),zi + gi) >_ di) < exp + P. N Now, apply Lemma 3. with Az = A~+Eallxll2 whenever x E zi4-ki and with S = U(zi+Ki), to get a lower estimate for Q But Q 4- ½exp(-e2a~/2) > Q + P(g > ae) -R:P( U [llxll=(~-"~g~)l/2 + < x'h> ~)'~]) xes -> P( N [lla(~)ll~ + Ilxll.,g ~ ~]) xes -> P( r] [~:~(~]~ gy) '/~ + (~,h} _> :,~]) - R. xes -< ~((~;)~/~ -< (~- ~)a~) 4- ~( U U i:lxgzi+ki N -< exp(-~4/2 + ~P( U i= i N <_ exp(--e2a~/2) 4- E P( U i= xezi+ki N [(x,h} _< Ai - ( - 2~)all~l[~]) [(x,h} >_ ( -- 2~)allxll2 - Ai]) [(~,h) > ( - 2~)~t, - ~,]).

17 00 Since the function f(h) = max (x,h}, defined for h E ~n, has Lipschitz constant Ti and, zezi+ki since E(f) = E(K/*) and, given the assumption ( - 2 )ati > Ai + E(K/*), it follows from Theorem 3.2 that P(f concludes the proof. >_ ( - 2 )ati -- Ai) _< exp ( -- (( -- 2e)ati -- Ai -- IE(K~))2/2T2), Q Taing ~ ~ oo, (I + ~)d~ J~ 0 for each i, we obtain the measure of all -codimensionai N subspaces which miss U(zi + Ki). Corollary 4.2. we have Under the assumptions c > 0 and ( - 2 )atl > E(K~. ) for all i =,...,N, "Y... (~... ; (n(z,+kd = forall i=i,...,n) N 3 > - ~ exp(-d4/2 ) - ~exp ( - {( - 2 )at, -- ~(K:)}~/2T,~). i=l 5. Random Subspaces which Escape through at least One Mesh Theorem.4 gave a sufficient condition for the existence of a subspace of given codimension which misses a given subset of the sphere. However, it may happen that every subspace of codimension hits the set and yet there might be a subspace which misses a "piece" of this set. To study such cases we consider a finite collection Al ( =, 2,..., L) of closed subsets of S~ - and as for a condition which implies that there exists a subspace of codimension which misses at least one of the sets At. To obtain this condition we shall use the following theorem proved in [G3], of which Theorem A is a special case. Theorem C. Let f(x) = minmaxminmax.., xi~,i2... i~ where for each ~ =,3,...,, the I $2 $3 index it ranges over some finite non-empty set C (il, i2,..., it-) (which, as indicated, shows that this set may depend on the previous choice of il, i2,.-., it-l). Given two distinct vector indices i = (Q,...,i) and j = (j,...,j) et m = m(i,j) be the first coordinate such that im# jm. ff {Xi} and {Yi} are two Gaussian processes such that (i) EtXi - Xjl ~ <_ lyi - YjI 2 if rn is even Oi ) EIX i _ Xjl2 >_ ElY ~ _ yj{2 if rn is odd, then El(X) <_ El(Y). Theorem A is obtained from Theorem C by taing -- 2 and _ il _< n, _< i2 _<: rn.

18 0 Corollary 5.. Let {AI}L=I be a finite collection of closed subsets of the sphere S2 t- and G(w) = E gij(w)ei e i a Gaussian operator from D:g n to ll:g. Then i= j=l /E(max{ ~ea,min IIC(~)(=)ll= + 2gt})_>a-a,~ f minmax<x,u)m._t(du).t xca, S~*- Proof." We apply Theorem C and for each g 6 {,2... L}, x E At and y 6 S - we define the two triple indexed Gaussian processes Yt,z,~ = E giyi + hixi and Xt,z,y : (G(w)x,y) + 2gt j=l i= For any two triples a = (~, x, y) and/9 = (g, x', y') we have and n mgo - g~l ~ = ~(yj - ~})~ + ~(~, - d) ~ = 4-2(~,,,~'/- ~(~,y')./= i= ~7,Xc, - Xel 2 : E(xiyi - xiyi)' "= + 8( - at,t,) = 0-86t,l, - 2<x,x')(y,y'). i=l j=l Therefore if ~ = g and x # x', i.e., when m(a, 8) = 2, then E[Y~ - Y~[= - ~;[x~ - xel 2 = 2( - <z,z'))( - (y,y')) >_ o. To chec condition (ii) for Theorem C, notice that if ~ = g and x = x' then/eixa - X~[ 2 = EiY~ - Yzl ~, and if ~ # ' then EIY~, - yzl ~ - EIx~ - Xzl ~ : -2(3 + <~, ~'> + <y,y') - <~, ~')<y, y'>) _< o, hence, by Theorem C,/Emin max min Xt z y < /Emin max min Yt,z,u therefore, replacing t xeat Y ES~-2 ' ' -- t xcat ycs 2~- X by -X and Y by -Y we obtain /~ maxmin max X~,z,y =/Emax{ min IIG(~)(~)II= + 2g~} t z y t xeat > E max min max Y,... E(Eg~)'/2+lEmtaxn~nEhixi -- t z y ' '~' =a-- Etnxax m ~. = --an minmax(x,u)mn_l(du). S~- g xeai n As a conclusion we obtain the following generalization of Theorem.4.

19 - seat - 02 Theorem l <<n Let {Ae}2=l be a finite collection of closed subsets of the sphere S~ -~ and If (5.) a >an / minmax(x,u}m,~_l(du)e zca, +c Ix/~L S~-- then there is a subspace of codimension which misses at least one of the sets Ae. Proof: Since/E max 2ge <_ c ~ we obtain from Corollary 5. t f /~max min tla(~)(x)ll~ >a--a, I minmax(x,u>mn_l(du)-c l/~l > 0. scat -- J xeat But, since the subspace E~ = G(w)-'(0) has codimension a.e., there is a subspace E~ o of codimension which misses at least one of the sets Ae. We now give two different sufficient conditions on the sets Ae which imply inequality (5.) and hence the conclusion of Theorem 5.2. Corollary 5.3. Let {Ae}e=l L be closed subsets of S~ -I, A = UAe and <_ < n. Assume one of the following two conditions are satisfied: L (a) there exists 0 < - < ~r such that N (Ae).~ = and = L u (5.2) a > an cos ~ -b c ~. (b) Let a, t3 (0 <_ a, fl <_ ~r) be defined by a = maxdiamat and t L fl=sup{t_>o; (-A c) N(Ae)t = } ~= L (withfl=oif(-a C) N At#C) and assume t=l (5.3) a > an { -- mr,- (A) cos a + ran- (A c) cos fl} + e lx/[o ~. Then inequality (5.) is satisfied. Proof: We shall estimate the integral which appears in (5.). If condition (a) holds, then for every u E S~ -, u ~ (Aeo)~ for some o, hence for every x E Aeo the geodesic distance p(x, u) > 7, therefore rnin max (x, u} < max (x, u) < cos -'/ t. zeat - o -

20 - xca, - t~fl 03 n-- which shows that (5.2) implies (5.). If condition (b) holds, let u E S 2 -u E A, so -u E A~ o for some t0. But then for all x E Aeo we have. If u E -A, then p(=, ~,) _> p(-u, u) - p(-u, z) z ~- -,~ hence min max(x, u) < max (x, u) _< cos(zr - a) = - cos a. zca~ zea~o On the other hand, let 0 < t < fl (tae t = 0 if fl = 0). If u E -A c, then u ~ (A~o)t for some go, so p(u, x) > t for every x E A~ o, hence minmax(x,u) < max(x,u) <cost t xea~ - o cosfl. Therefore, / = /+ / <---mn-l(-a)cosa+mn-l(-ac)cost ~ S~ -~ -A --A c = -m~._,(a)cos,~ +.~._I(A ~) cos~. This proves that (5.3) implies (5.). n Remar (5.). Corollary 5.3(a) is illustrated by the following example. Let 0 < E,& <, ---- v/~n -(-*)/4 and L <_ exp(el&2n e) be an integer. If {At}L_I is a collection of closed L subsets of S~ - such that N (Al)~ =, then there exists a subspace of dimension rn >_ t=l 5n (~+)/2 in J~:~'~ which misses at least one At, provided n >_ N(e). (cl denotes a positive absolute constant.) The proof of this is done by estimating the quantities which appear in inequality 5.2(a). Tae ~ n - 3rt (~+)/2, so a ~ /2 ~ hi/2( - ½5n(~-)/2), an ~ cos'r ~ - q2/2 = - *n -(-*)/2 (since y constant. r/./2,, 0) and choose cl to be a suitable positive r,~oo As done in section 3, we can assign the probability measure "/,~,r,- to the set of all subspaces ~ E ~... which miss at least one of the sets At ( < ~ < ). For this estimate we need the following theorem of [G3] which extends Theorem B above. Theorem D. Let f(x) and the processes {Xi), {Y~} be as in Theorem C and in addition to conditions (i) and (ii) we assume also tfiat EX 2 = EYi 2 for all i. Then for all choices of the real sequence {A,}, P(NUNU.tx,-> -< P(NUNU -> il lit i3 ij. i~ i3

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