Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

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1 Caad. J. Math. Vol. 60 (1), 2008 pp Covex Bodies of Miimal Volume, Surface Area ad Mea Width with Respect to Thi Shells Károly Böröczky, Károly J. Böröczky, Carste Schütt, ad Gergely Witsche Abstract. Give r > 1, we cosider covex bodies i E which cotai a fixed uit ball, ad whose extreme poits are of distace at least r from the cetre of the uit ball, ad we ivestigate how well these covex bodies approximate the uit ball i terms of volume, surface area ad mea width. As r teds to oe, we prove asymptotic formulae for the error of the approximatio, ad provide good estimates o the ivolved costats depedig o the dimesio. 1 Notatio Let us itroduce the otatio used throughout the paper. For ay otios related to covexity i this paper, cosult R. Scheider [19]. We write o to deote the origi i E,, to deote the scalar product, ad to deote the correspodig Euclidea orm. Moreover for o-colliear poits u, v, w, the agle of the half lies vu ad vw is deoted by uvw. Give a set X E, the affie hull, the covex hull ad the iterior of X are deoted by aff X, cov X ad it X, respectively. I additio if X is covex, the the relative boudary ad the relative iterior of X with respect to aff X are deoted by X ad relit X, respectively. We write B to deote the uit ball cetred at o, ad S 1 to deote B. The k-dimesioal Hausdorff measure is deoted by H k (see [8, 16] for defiitio ad mai properties). We ormalize it i a way such that it coicides with the Lebesgue measure i E k. If M is a bouded measurable subset of E, the we call H (M) the volume V (M) of M. As usual, we call a compact covex set i E with o-empty iterior a covex body, ad a two-dimesioal compact covex set a covex disc. For a covex body C, we write S(C) = H 1 ( C) to deote its surface area. Whe itegratig o C, we always do it with respect to H 1. The two-dimesioal Hausdorff measure of a two-dimesioal covex compact set, or of a measurable subset X of the boudary of some covex body i E 3, is also called the area A(X) of X. We recall that x is a extreme poit of a covex compact set C if x does ot lie i the relative iterior of ay segmet cotaied i C. We write extc to deote the family of extreme poits, ad ote that extc forms the miimal set whose covex hull is C. Received by the editors May 30, 2005; revised August 27, The first author was supported by OTKA grats T ad The secod author was supported by OTKA grats T , ad , ad by the Marie Curie TOK project DiscCovGeo. The third author was supported by the Marie Curie Research Traiig Network Ph.D. AMS subject classificatio: Primary: 52A27; secodary: 52A40. c Caadia Mathematical Society

2 4 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche Give a compact covex set C i E, its support fuctio h C (u), u E, is defied by h C (u) = max x C x, u. I particular, for ay u S 1, the width of C i the directio u is h C (u) + h C ( u). Therefore the mea width of C is M(C) = 2 S(B ) S 1 h C (u) du. I particular M(B ) = 2, ad if C is a covex disc, the M(C) = 1 π S(C) accordig to the Cauchy formula (see [19]). We ote that the volume, surface area, ad the mea width of a compact covex set C i E ca be expressed as the mixed volumes (quermassitegrals or ormalized itrisic volumes), (1.1) V (C) = V (C,..., C), S(C) = V (C,..., C, B ), M(C) = 2 V (B ), V (C, B,...,B ). 2 Itroductio Let us defie the mai objects of study i this paper. Defiitio Give r > 1, we write F r to deote the family of covex bodies i E which cotai B, ad whose extreme poits are of distace at least r from o. Moreover let P r, Q r ad W r be elemets of F r with miimal volume, surface area, ad mea width, respectively. The miima do exist accordig to the Blaschke selectio theorem, ad all extreme poits of Pr, Q r ad Wr lie o rs 1 by the mootoicity of the volume, surface area ad the mea width. Ufortuately, we do ot kow whether the extremal covex bodies are polytopes. For example, if r is reasoably large, the it is cojectured that all extremal covex bodies are right cyliders whose bases are uit ( 1)-balls. Aswerig a questio of J. Molár [15], K. Böröczky ad K. Böröczky, Jr. [3] proved that Pr 3 ad Q 3 r are regular octahedra whe r = 3, ad are regular icosahedra whe r = As discussed i [3], o regular polytope is extremal i its class if 8. Therefore i this paper we cosider the case whe r teds to 1 ad there is o restrictio o the dimesio. Give real fuctios f (r) ad g(r) of r > 1, we write f (r) f (r) g(r) if lim r 1 g(r) = 1. I additio we write g(r) = O( f (r)) if g(r) c f (r) for some costat c depedig oly o. Theorem 2.1 If 2 ad r > 1 teds to 1, the V (Pr \B ) θ V () (r 1), S(Q r ) S(B ) θ S () (r 1), M(W r ) M(B ) θ M () (r 1), where θ V (), θ S (), ad θ M () are positive costats depedig oly o.

3 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 5 Sice all V (rb ) V (B ), S(rB ) S(B ), ad M(rB ) M(B ) are of order r 1 if r is close to 1, it is ot surprisig that we have the factor r 1 i Theorem 2.1. We ote that Theorem 2.1 is proved i [4] if 3. Additioally, it was determied i [4] that i the plaar case, ad θ V (2) = 2π 3, θ S(2) = 4 3, θ M(2) = 4π 3 θ V (3) = π, θ S (3) = 3π, θ M (3) = 7 6 i the three-dimesioal case. Moreover [4] proved that if r is close to 1, the the typical faces of P 3 r, Q 3 r, ad W 3 r are asymptotically regular triagles. For large, combiig Theorem 2.1 ad Lemma 5.1 yields that there exist positive absolute costats c 1 ad c 2 satisfyig (2.1) (2.2) (2.3) c 1 S(B ) < θ V () < c 2 l S(B ), c 1 S(B ) < θ S () < c 2 l S(B ), c 1 < θ M() < c 2 l, where S(B ) = π 2 Γ( 2 + 1) (see [19]). Next we state a theorem that is essetial i provig (2.1), (2.2), ad (2.3). Theorem 2.2 the If ρ > 0 ad C is a covex body i E with x ρ for all x extc, C x 2 dx ρ2 9 V (C). Remark Theorem 2.2 is optimal up to a absolute costat factor, as is show by the example of regular simplices iscribed ito ρ B. A field closely related to our paper is polytopal approximatio where a give smooth covex body C i E is approximated by polytopes of restricted umber of vertices ad facets. A typical problem is to cosider the iscribed polytopes P k,v ad P k,m with at most k vertices of maximal volume ad of maximal mea width, respectively. As k teds to ifiity, asymptotic formulae expressig V (C) V (P k,v ) ad M(C) M(P k,m ) are kow. I additio, the iscribed polytope P k,h with at most k vertices ad miimizig the so-called Hausdorff distace from C (see (3.1)) is well ivestigated. It is also kow that if C is a ball i E 3, the the typical faces of the extremal polytopes are asymptotically regular triagles. For refereces, see the ice surveys by P. M. Gruber [12 14], ad the recet mauscript by K. J. Böröczky, P. Tick,

4 6 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche ad G. Witsche [6]. Various methods i this paper come from the field of polytopal approximatio, i which small parts of S 1 are approximated by paraboloids. The paper is structured i the followig way: Sectio 3 discusses polytopal approximatio from our poit of view, ad Sectio 4 proves Theorem 2.2. Sectio 5 presets the basic statemets ad mai idea of the proof for Theorem 2.1, ad it proves the approximate versio Lemma 5.1. The proof of Theorem 2.1 i the cases of volume ad surface area i Sectio 9 is based o the properties of covex hypersurfaces discussed i Sectios 6 ad 7, ad o Lemma 8.1 i Sectio 8 which describes how to trasfer polytopal approximatio ito itegratio i E 1. Theorem 2.1 i the case of the mea width is proved i Sectios 10 ad 11. We ote that the case of mea width is substatially easier tha the cases of volume or surface area. 3 Hausdorff Distace ad Polytopal Approximatio We will frequetly approximate covex bodies by polytopes (see [11 13] for geeral surveys). A atural measure of closeess betwee compact sets is the so-called Hausdorff distace. For a x E ad a compact X E, we write d(x, X) to deote the miimal distace betwee x ad the poits of X. If K ad C are compact sets i E, the their Hausdorff distace is (3.1) δ H (K, C) = max { max d(x, C), max d(y, K)}. x K y C I the case whe C ad K are covex, the maximum of d(x, C) amog x K is attaied at some extreme poit of K. We always cosider the space of compact sets as the metric space iduced by the Hausdorff distace that is readily a metric. I particular we say that a sequece {K m } of compact sets teds to a compact set C if lim m δ H (K m, C) = 0, ad clearly C is covex if every K m is covex. For the mai properties of the Hausdorff distace, see [19]. For example, the Blaschke selectio theorem says that if {K m } is a sequece of compact covex sets that are cotaied i a fixed ball, the {K m } has a subsequece {K m } that teds to some compact covex set C. I additio the volume, surface area ad the mea width are cotiuous fuctios of covex bodies. This latter property follows from the followig fact: if the covex bodies K ad C cotai B, the [1 + δ H (K, C)] 1 K C [1 + δ H (K, C)] K. Let F r deote the family of all C F r satisfyig extc rs 1. The P r, Q r, W r F r. Lemma 3.1 Let 1 < r < 2 ad let 0 < µ < 1 4 r 1. If C F r, the there exists a polytope M F r such that the distace betwee ay two vertices of M is at least µ, ad M satisfies δ H (M, C) 4µ r 1. Proof Let x 1,...,x k be a maximal system of poits of rs 1 such that d(x i, x j ) µ for i j. Now the vertices of M are the x i s whose distace from some extreme poit of C is at most 2µ.

5 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 7 First we show that M cotais B. Let H + be ay closed half space that avoids B, ad whose boudig hyperplae touches B. The H + cotais some extreme poit y of C, hece there exists a z rs 1 of distace at most µ from y satisfyig that rs 1 (z + µb ) H +. Sice z + µb cotais some x i that is the of distace at most 2µ from y, we coclude B M. Next we estimate the Hausdorff distace. If y is a extreme poit of C, the its distace from some vertex x i of M is at most 2µ, hece d(y, M) d(y, cov{x i, B r2 1 }) 2µ. r The aalogous argumet for d(x i, C) where x i is ay vertex of M completes the proof of Lemma 3.1. Let us remark that a result of R. Scheider [18] about approximatio of smooth covex bodies by iscribed polytopes of restricted umber of vertices with respect to the Hausdorff distace yields the followig statemet. If N(r) is the miimal umber of vertices of polytopes i F r, the N(r) c() (r 1) 1 2 where c() is a explicit costat depedig o. 4 Proof of Theorem 2.2 The proof of Theorem 2.2 will use Lemma 4.3, whose proof i tur is prepared by verifyig Propositio 4.1. Propositio 4.1 Let z 1,...,z +1 be the vertices of a regular simplex i E with z i = 1, i = 1,..., + 1, let e 1,...,e be a orthoormal basis i E, ad let a 1,...,a R. The there exists i 0 such that a 2 j z i0, e j 2 1 j=1 a 2 j. j=1 Proof We thik that E is embedded ito E +1 as a liear subspace, ad let y be oe of the uit ormals to E i E +1. I particular, y i = z i + y, + 1 i = 1,..., + 1, form a orthoormal basis of E +1. For ay e j, we have e j, y i = +1 e j, z i for i = 1,..., + 1, hece +1 1 = e j = y i, e j 2 = +1 z i, e j

6 8 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche It follows that +1 a 2 j z i, e j 2 = j=1 +1 a 2 j j=1 z i, e j 2 = + 1 a 2 j, j=1 which i tur yields the existece of z i0. For a liear map A, we recall a geeral fact that follows easily from the pricipal axis theorem applied to A t A: Fact 4.2 (Polar decompositio) Let A: E E be a liear map. The there are orthogoal maps U, V : E E ad a diagoal map D: E E with diagoal elemets a 1 a 2 a 0 such that A = U DV. The diagoal elemets of D are uique ad called the sigular umbers of A. We recall that the cetroid of a bouded measurable set M i E is the poit c = 1 x dx, V (M) M ad for ay y E, we have (4.1) M x c, y dx = 0. We ote that if S = cov{x 1,...,x +1 } ad T = cov{z 1,...,z +1 } are simplices i E whose cetroids are the origi o, the +1 x i = o = +1 z i, hece there exists a uique liear map A with A(z i ) = x i, i = 1,..., + 1. Lemma 4.3 For r > 0, let S = cov{x 1,...,x +1 } be a simplex i E with x i r, i = 1,..., + 1 ad with its cetroid at the origi o, ad let T = cov{z 1,...,z +1 } be a regular simplex i E with z i = 1, i = 1,..., + 1. If A is the liear map with A(z i ) = x i, i = 1,..., + 1, ad a 1,...,a are the sigular umbers of A, the a 2 j r 2. j=1 Proof Applyig orthogoal trasformatios to S ad T chages either the coditios o S ad T or the sigular umbers of A, hece we may assume that A = D where D is the diagoal matrix with diagoal elemets a 1,...,a. Let e 1,...,e be the correspodig orthoormal basis i E. By Propositio 4.1 there exists i 0 such that 1 a 2 j j=1 a 2 j z i0, e j 2 = j=1 Dz i0, e j 2 = Dz i0 2 = x i0 2 r 2. j=1

7 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 9 Let T be a regular simplex whose cetroid is the origi. Sice the positive defiite quadratic form q T (u) = T x, u 2 dx is ivariat uder the symmetries of T, we deduce that q T (u) = λ u, u for suitable λ > 0 depedig o T, amely, T is i isotropic positio (see [10]). I particular if e 1,...,e form a orthoormal basis of E, the T x, e i 2 dx = M x, e j 2 dx for i j, therefore (4.2) T x, e i 2 dx = 1 T x 2 dx, i = 1,...,. Proof of Theorem 2.2 We may assume that ρ = 1, ad by approximatio also that C is a polytope. Subdividig C ito simplices shows that it is sufficiet to prove Theorem 2.2 for a -simplex S = cov{x 1,...,x +1 } with x i 1, i = 1,..., + 1. We write c to deote the cetroid of S, ad we have x 2 dx = x + c 2 dx = x 2 dx + 2 x, c dx + c 2 dx. S c S S c Sice o is the cetroid of S c, (4.1) yields x, c dx = 0, hece S c x 2 dx = x 2 dx + c 2 dx c 2 V (S). S c S S c S c Now if c 2 1 4, the Theorem 2.2 readily follows. Therefore to prove Theorem 2.2, it is sufficiet to verify that if c 2 1 4, the (4.3) x 2 dx 1 9 V (S). S c It follows by the triagle iequality that the vertices x i c, i = 1,..., + 1 of S c satisfy x i c Let T = cov{z 1,...,z +1 } be a regular simplex with z i = 1 ad let A be the liear map defied by A(z i ) = x i c. We fix a orthoormal basis e 1,...,e of E. Possibly after applyig a orthogoal trasformatio to T, we may assume that the polar decompositio of A is of the form A = U D, where U is a orthogoal trasformatio, ad D is a diagoal map whose diagoal elemets are the sigular umbers a 1,...,a of A. After substitutig x = U y, we obtai S c x 2 dx = DT Next the substitutio y = Dw leads to S c x 2 dx = T y 2 dy = S c DT Dw, e i 2 det(d) dw = det(d) y, e i 2 dy. a 2 i w, e i 2 dw. T

8 10 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche Sice T is i isotropic positio (see (4.2)), we deduce by ( ) > 1 4 ad by Lemma 4.3 that S c x 2 dx = det(d) w 2 dw T a 2 i det(d) 4 w 2 dw. T I order to estimate T w 2 dw from below, we recall the Stirlig formula i the form (see [1]) t t t t 2π t < Γ(t + 1) < 2π(t + 1), e t e t which i tur yields V (T) = V (B ) = + 1 Γ( + 1) ( + 1 ) /2 3 > 2 e π/2 Γ( 2 + 1) < 1 e/2 π / /2 2π It follows by cosiderig the part of T outside e π B that T w 2 dw Therefore we coclude (4.3) by S c 1 2π ; ( 1 2 ) e 3 2 π V (T) > 4 9 V (T). x 2 dx det D 9 V (T) = 1 9 V (S), which i tur completes the proof of Theorem Some Prelimiary Observatios Cocerig Theorem 2.1 We assume the dimesio satisfies 3 for the whole sectio. The aim of the sectio is first to outlie the basic idea of the proof of Theorem 2.1, ad the to provide a raw form (see Lemma 5.1). Fially we will prove Lemma 5.2, which helps to fid a suitable cogruet copy of a give patch o S 1. Whe determiig the asymptotics of the volume, surface, ad mea width differece, we will replace the optimal covex bodies i Fr by polytopes with the help of Lemma 3.1. After fixig a small ε > 0, we eed estimates up to a factor 1 + O(ε) for ay r > 1 very close to 1. Sice ay facet of the extremal bodies is of diameter at most 2 r 2 1, we will cosider patches of size r 1 ε. A very useful property of cofiguratios i E 1 is that they ca be dilated, hece we trasfer the itegrals over S 1 to itegrals over E 1. I additio, i the cases of volume ad surface area, we substitute the patches o S 1 by patches o paraboloids because paraboloids better suit dilatio i E 1.

9 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 11 I this sectio we prove two auxiliary statemets, Lemma 5.1, which is a raw form of Theorem 2.1, ad Lemma 5.2, which allows choosig suitable patches o S 1. Let r (1, 2). We write π S 1 to deote the radial projectio ito S 1, hece if F rb is a compact covex set with aff F it B =, the for ay x, y F, π S 1(x) π S 1(y) x y r 2 π S 1(x) π S 1(y). Give a polytope P F r, let F 1,...,F k be the facets of P. For i = 1,...,k, we write x i S 1 to deote the uit exterior ormal to F i, ad ν i to deote the distace of aff F i ad B, moreover we defie z i = (1 + ν i )x i aff F i. If y F i ad x = π S 1(y), the y x = 1+ν i x,x i 1, therefore the formula (6.3) proved by J. R. Sagwie- Yager [17] with X = S 1 ad Y = P yields (5.1) V (P) V (B ) = 1 = k F i π S 1 (F i ) (1 + ν i ) x, x i 1 dx k ( 1 ) 2 x z i 2 + ν i dx + O((r 1) 2 ). Cocerig the mea width, let v 1,...,v l S 1 be the poits such that rv 1,...,rv l are the vertices of P. We write Q to deote the polytope determied by the taget hyperplaes at v 1,...,v l S 1, ad G j to deote the facet of Q cotaiig v j for j = 1,...,l. Thus (5.2) M(P) M(B ) = 2 S(B ) l j=1 π S 1 (G j ) = 2(r 1) 1 S(B ) x, rv j 1 dx l j=1 π S 1 (G j ) x v j 2 dx + O((r 1) 2 ). Let us show that the orders of V (P r ) V (B ), S(Q r ) S(B ), ad M(W r ) M(B ) are all r 1. Lemma 5.1 If 1 < r < r 0, the c 1 S(B )(r 1) < V (P r ) V (B ) < c 2 l S(B )(r 1), c 1 S(B )(r 1) < S(Q r ) S(B ) < c 2 l S(B )(r 1), c 1 (r 1) < M(W r ) M(B ) < c 2 l where c 1, c 2 > 0 are absolute costats, ad r 0 > 1 depeds o. (r 1),

10 12 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche Proof To prove the upper bouds, we start with the mea width. Sice M(W r ) M(rB ) = M(B ) + 2(r 1), we may assume that is large. For v S 1 ad ϕ (0, π/2), we defie B(v, ϕ) = {x S 1 : v, x cosϕ}. Projectig orthogoally to the taget hyperplae at v shows that (5.3) H 1 (B 1 ) si 1 ϕ < H 1 (B(v, ϕ)) < H 1 (B 1 ) si 1 ϕ cos ϕ. Let ψ = arccos 1/r. Accordig to K. Böröczky, Jr. ad G. Witsche [7], there exists a coverig of S 1 by spherical balls B(v 1, ψ),...,b(v l, ψ) such that (5.4) l H 1 (B(v j, ψ)) < 400 l S(B ) < 2 S(B ). j=1 Let P be the covex hull of rv 1,...,rv l. Sice for ay x S 1 there exists v j with rv j, x 1, we deduce that B P, hece P Fr. I the followig we use the otatio of (5.2), ad defie Ω = j=1,...,l ( ( B v j, 1 4 l ) ) ψ. Sice (1 4 l ) 1 < 2/ 4 for large, we deduce by (5.3) ad (5.4) that if r is close to 1, the (5.5) H 1 (Ω) < 3 2 S(B ). We have ψ 2(r 1). Therefore if r is close to 1, the [ 1 cos (1 ( 4 l ) ] ( ψ ( I particular if x π S 1(G j )\Ω, the ( x v j l ) 1 4 l 1 [( 2 ) 4(r 1). 1 4 l ) 4(r ( 1) l ) ] 2 ψ 32 l ) (r 1). Therefore we coclude by (5.2) ad (5.5) that if r is close to 1, the ( M(P) M(B ) < 2(r 1) 1 3 )( l ) (r 1) + 1 (r 1). I particular, if is large eough ad r (1, r 0 ) for suitable r 0 > 1 depedig o, the M(P) M(B 33 l ) < (r 1).

11 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 13 This settles the case of the mea width. I additio, the upper bouds of Lemma 5.1 i the cases of surface area ad volume follow from the cosequeces S(P) ( M(P) ) 1 S(B ) V (P) ( M(P) ) M(B ad ) V (B ) M(B ) of the Alexader Fechel iequality for mixed volumes (see (1.1) ad [19]). To prove the lower bouds, we first cosider the case of the volume. Accordig to Lemma 3.1, it is sufficiet to prove the lower boud for ay polytope P Fr with ext P rs 1 where we use the otatio of (5.1) for P. It is eough to show that for each F i, (5.6) F i ( 1 2 x z i 2 + ν i ) dx > c H 1 (π S 1(F i )) (r 1) where c is a positive absolute costat. If ν i, the (5.6) readily holds. Otherwise aff F i itersects B i a ( 1)-ball B i of radius larger tha r 1. Sice the vertices of F i lie o B i, Theorem 2.2 completes the proof of (5.6), ad i tur of the lower boud i Lemma 5.1 i the case of the volume. Fially the cases of surface area ad the mea width follow from the Alexader Fechel iequality for mixed volumes (see (1.1) ad [19]) i the form S(Q r ) ( V (Q S(B ) r ) ) 1 V (B ) ad r 1 ad the iequalities V (Q r ) V (P r ) ad V (W r ) V (P r ). M(W r ) ( V (W M(B ) r ) ) 1 V (B, ) A essetial step of the argumets for all the three quermassitegrals is to fid the right copy of a give patch o S 1. Let us recall that SO() deotes the group of orietatio preservig orthogoal trasformatios of E (see [19]). Lemma 5.2 If f is a bouded measurable fuctio o S 1 ad X S 1 is measurable with H 1 (X) > 0, the there exist g 1, g 2 SO() such that f (x) dx H 1 (X) S(B f (x) dx f (x) dx. ) g 1 X Proof We write µ to deote the (ivariat) Haar measure o SO() ormalized i a way such that µ 1 (SO(1)) = 2π, ad for ay measurable Z S 1 ad x S 1, S 1 µ {g SO() : g 1 x Z} = µ 1 (SO( 1)) H 1 (Z). I additio we write χ Z to deote the characteristic fuctio of a set Z S 1. For g SO(), we defie h(g) = gx f (x) dx = χ S 1 X (g 1 x) f (x) dx. It follows by the Fubii theorem that h(g) dµ (g) = χ X (g 1 x) f (x) dµ (g)dx SO() S 1 SO() g 2 X = µ 1 (SO( 1)) H 1 (X) S 1 f (x) dx.

12 14 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche Therefore there exist g 1, g 2 SO() satisfyig h(g 1 ) H 1 (X) H 1 (S 1 ) f (x) dx h(g 2 ). S 1 6 Covex Hyper Surfaces We will cosider patches o the boudary of covex bodies. We say that a X E is a covex hypersurface if cov X is closed with o-empty iterior ad cotais X i its boudary, ad X is the closure of its relative iterior with respect to the boudary of cov X. Moreover, the relative boudary relbd X of X is of ( 1)-measure zero. We write relit X to deote the relative iterior of X, ad u X (x) to deote some exterior uit ormal at x relit X. We ote that u X (x) is uique for all x relit X but of a set of ( 1)-measure zero. Whe itegratig over X, we always do it with respect to H 1 ( ). If the closest poit x of cov X to some y lies i X, the we write π X (y) = x. We ote that if π X (y) ad π X (y ) are well defied, the (6.1) π X (y) π X (y ) y y. If the covex hypersurface Y E is the uio of F 1,...,F k such that each F i is a Jorda measurable subset of some hyperplae ad has positive ( 1)-measure, ad aff F 1,...,aff F k are pairwise differet, the we say that a Y is a covex piecewise liear hypersurface, ad call F 1,...,F k the facets of Y. For certai calculatios it is useful to cosider patches as graphs of fuctios. We thik E as E 1 R where x = (y, t) is the poit of E correspodig to y E 1 ad t R, ad defie B 1 = B E 1. If Ψ E 1 has o-empty iterior i E 1, ad θ : Ψ R is ay fuctio, the the graph of θ is Γ(θ) = {(y, θ(y)) : y Ψ} E. I particular if Ψ ad θ are covex, the Γ(θ) is a covex hypersurface. We say that a covex hypersurface X is C 2 if ay poit of X has a relatively ope eighbourhood o X that is cogruet to the graph of some C 2 fuctio. I order to defie the priciple curvatures at x 0 relit X, we may assume that E 1 is the taget hyperplae to X at x 0 = (y 0, 0), ad a eighbourhood X 0 X of x 0 is the graph of a C 2 fuctio θ o a ope covex Ψ E 1. The the priciple curvatures κ 1 (x 0 ),...,κ 1 (x 0 ) of X at x 0 are the eigevalues of the symmetric matrix correspodig to the quadratic form represetig the secod derivative of θ at y 0. For x X, we defieσ 0 (x) = 1, ad write σ j (x) to deote the j-th symmetric polyomial of the pricipal curvatures for j = 1,..., 1; amely, σ j (x) = κ i1 (x) κ i j (x). 1 i 1 < <i j 1 For the rest of the sectio, let X be a covex C 2 hypersurface, ad let Y be a covex hypersurface such that π X is defied o Y ad is ijective with X = π X (Y ). Moreover, there exists η > 0 such that (6.2) u X (π X (y)), u Y (y) η for ay y relit Y.

13 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 15 It follows by (6.1) that π X (Y ) is also a covex hypersurface with H 1 (π X (Y )) H 1 (Y ). I additio if Z relit π X (Y ) is a covex hypersurface, the the subset Z of Y satisfyig π X (Z ) = Z is a covex hypersurface by (6.2). If π X (y) = x for y relit Y, the we write y = x Y ad defie r X,Y (x) = y x. We defie Ω(X, Y ) to be the uio of all segmets cov{y, π X (y)} for y Y, which satisfies (6.3) V (Ω(X, Y )) = j=1 1 r X,Y (x) j σ j 1 (x) dx j X accordig to J. R. Sagwie-Yager [17]. I additio the method of K. Böröczky, Jr. ad M. Reitzer [5] yields the followig formula for the differece of the ( 1)- measure of patches. Lemma 6.1 Usig the otatio as above, ( H 1 (Y ) H 1 1 ) (X) = u X (x), u Y (x Y ) 1 dx X 1 + j=1 X r X,Y (x) j σ j (x) u X (x), u Y (x Y ) dx. Proof For small µ > 0, we write Ω µ to deote the family of poits z E such that the closest poit of cov Y to z lies i relit Y, ad π Y (z) z µ. Next let X µ be the family of poits x X with d(x, relbd X) 2µ, ad let Y µ Y satisfy π X (Y µ ) = X µ. For ay x X µ, there exists a uique boudary poit z Ω µ with d(z, Y ) = µ ad π X (z) = x, ad we write Z µ to deote the family of all such z as x rus through X µ. Now relbd X µ might be positive for some but oly a coutable family {µ i } of µ > 0. Therefore X µ ad Z µ are covex hypersurfaces for µ > 0, µ {µ i }, with π X (Z µ ) = X µ. I additio if x Y Y is a smooth poit of Y for x X µ, the r Xµ,Z µ (x) r X,Y (x) + µ η (cf. (6.2)), ad r Xµ,Z µ (x) = r X,Y (x) + µ + o(µ) as µ teds to zero. x, x Y Sice the π X image of sigular poits of Y are of ( 1)-measure zero, we deduce by (6.3) ad as X ad Y are Jorda measurable that H 1 V (Ω µ ) (Y ) = lim µ 0 µ µ {µ i } = j=1 I tur we coclude Lemma 6.1. X V (Ω(X µ, Z µ )) V (Ω(X µ, Y µ )) = lim µ 0 µ µ {µ i } r X,Y (x) j 1 σ j 1(x) x, x Y dx.

14 16 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche 7 Near Spherical Covex Hyper Surfaces For ε (0, 1 16 ), let ρ (0, ε2 ), ad let Ψ εb 1 be a ( 1)-dimesioal covex body with o relit Ψ. I additio let θ be a o-egative C 2 fuctio defied o Ψ such that writig l y to deote the liear form represetig the derivative of θ, ad q y to deote the quadratic form represetig the secod derivative of θ at y Ψ, we have θ(o) = 0, l o (z) = 0, ad z 2 q y (z) (1 + ε) z 2 for z E 1. We defie X = Γ(θ), ad write κ 1 (x),..., κ 1 (x) to deote the priciple curvatures at x relit X. We ote that if y Ψ ad x = (y, θ(y)), the (7.1) u X (x) = (1 + l y 2 ) 1/2 (l y, 1). We deduce, by the Taylor formula for y, z Ψ, x = (y, θ(y)), (7.2) (7.3) θ(z) θ(y) l y (z y) = 1 2 q y+t(z y)(z y) for t (0, 1), = 1 2 z y 2 + O(ε) y z 2, l z l y = z y + O(ε) z y, κ i (x) = 1 + O(ε), i = 1,..., 1. Now for ay x X, X ca be thought as the graph of a suitable C 2 fuctio defied o the taget hyperplae at x, hece the discussio above ad (7.1) show that if x, x relit X, the (7.4) u X (x), u X (x ) = x x 2 + O(ε) x x 2. Next let X X be a covex hypersurface such that d(x, relbd X ) 4 ρ for x X. I additio let Y be a covex hypersurface such that π X is defied o Y ad is ijective with X = π X (Y ), ad if y relit Y u X (π X (y)), u Y (y) > 0. Therefore we may use the otatio of Sectio 6. I particular we assume that (7.5) r X,Y (x) 2ρ for x relit X. Naturally (6.3) ad Lemma 6.1 are very geeral, ad we provide three types of estimates based o them which will be useful i the later part of the paper. We write ξ = (o, 1) to deote the dowwards uit ormal to E 1. Sice all eigevalues of q y are at most 2 for ay y Ψ, there is a ball of radius 1/2 touchig X from iside at ay x X such that the ball itersects X oly i x. It follows by (7.5) that (7.6) u X (x), u Y (x Y ) ρ, which i tur yields (7.7) ξ, u X (x) = 1 + O(ε) for x relit X.

15 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 17 The first type of estimate is a rather rough oe; amely, (7.3), (7.5) ad (7.6) imply (7.8) (7.9) V (Ω(X, Y )) = O(ρ) H 1 (X), H 1 (Y ) H 1 (X) = O(ρ) H 1 (X). The secod type of estimate is eeded whe Y is a covex piecewise liear hypersurface. We write F 1,...,F k to deote the facets of Y, ad v 1,...,v k to deote the correspodig exterior uit ormals. We assume that for i = 1,...,k, v i = u X (x i ) for some x i X, ad x i + ν i v i aff F i for some ν i 0. Sice there exists a ball of radius 2 that touches X at x i ad cotais X if x π X (F i ), the the coditio r X,Y (x) 2ρ yields that x x i 4 ρ, hece r X,Y (x) = ν i x x i 2 + O(ερ) u X (x), u Y (x Y ) 1 = u X (x), u X (x i ) 1 = x x i 2 + O(ερ). We coclude by (6.3) ad Lemma 6.1 that (7.10) V (Ω(X, Y )) = H 1 (Y ) H 1 (X) = k π X (F i ) ( ν i x x i 2) dx + O(ερ)H 1 (X), k π X (F i ) ( ( 1)νi + 2 x x i 2) dx + O(ερ)H 1 (X). Fially Lemma 7.2 provides the third type of estimate, which allows us to shift betwee patches o spheres ad o paraboloids. Its proof uses the followig statemet. Propositio 7.1 Let z 1, z 2 E 1 such that z 2 z 1 τ for some τ > 0, ad let Y be the graph of a covex positive fuctio o z 1 + 2τ B 1 such that u Y (y), ξ 3/2 for y Y where ξ = (o, 1) as above. If y 1, y 2 Y satisfy that z i y i z i y i, ξ 3/2 for i = 1, 2 the y 1 y 2 2 [ z 1 z 2 + z 1 y 1 (z 1 y 1, o, z 2 y 2 )]. Proof We defie y 1 Y by the property that the vectors z 1 y 1 ad z 2 y 2 are parallel, ad prove (7.11) y 1 y 1 2 z 1 y 1 si (y 1, z 1, y 1). Let σ be the arc that is the itersectio of the triagle y 1 z 1 y 1 ad Y, ad let y be the poit of σ farthest from the segmet y 1 y 1. The the taget lie to σ at y is parallel to the lie y 1 y 1, hece u Y (y), ξ 3/2 yields that the agle of y 1 y 1 ad ξ is

16 18 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche betwee π 3 ad 2π 3. Thus the agle of the triagle z 1 y 1 y 1 at y 1 is betwee π 6 ad 5π 6, therefore the law of sies implies (7.11). Now a argumet as above shows that y 2 y 1 2 z 2 z 1, which i tur yields Propositio 7.1 by (7.11). Lemma 7.2 Give ε (0, ε 0 ) ad ρ (0, ε 8 ) where ε 0 (0, 1 16 ) depeds oly o, let the covex fuctios h, f 1, f 2 o (20 ρ) ε B 1 satisfy that f 2 (o) = 0, f 2 (o) = 0, f 1 ad f 2 are C 2, ad if y (3 ρ) ε B 1. The o the oe had, h(y) f 1 (y) f 2 (y) h(y) + 2ρ ad f 1 (y) 0, ad o the other had, writig q i,y to deote the quadratic form represetig the secod derivative of f i at y for i = 1, 2, we have z 2 q i,y (z) (1 + ε 8 ) z 2 for z E 1. We defie Y = Γ(h) ad X i = Γ( f i ), i = 1, 2 (see Figure 1). For a compact covex C E 1 satisfyig ρ 4ε B 1 C 2 ρ ε B 1 ad for i = 1, 2, we write X i = π Xi (C) ad Y i to deote the subset of Y satisfyig X i = π Xi (Y i ). The (7.12) (7.13) (7.14) H 1 ( X i ) = H 1 (C) + O(ε) H 1 (C) for i = 1, 2, H 1 (Y 1 ) H 1 ( X 1 ) = H 1 (Y 2 ) H 1 ( X 2 ) + O(ερ)H 1 (C), V (Ω( X 1, Y 1 )) = V (Ω( X 2, Y 2 )) + O(ερ) H 1 (C). X 2 X 1 Y X 2 X 1 E 1 C Figure 1

17 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 19 Proof It follows by (7.7) that if ε 0 is sufficietly small, the u Xi (x), ξ 3 2 for ay x relit X i. I additio if y = (z, h(z)) for z C ad u is a exterior uit ormal to Y at y, the d(y, X 1 ) ρ ad (7.2) yield that there exists x X 1 (y + 4 ωρ B d ) with u = u X1 (x), hece u, ξ 3 2, as well. I additio the coditios o h, f 1, f 2 ad applyig (7.2) to f 1, f 2 yield that (7.15) (7.16) (7.17) h(z) > 0 if z C\( 1 2 C), f 2 (z) 4ρ ε 2 if z C, f 2 (z) f 1 (z) 4ε 6 ρ if z C. Therefore combiig (7.9), (7.16), ad ρ/ε 2 < ε leads to (7.12). Moreover, writig γ 1(z) = X 1 cov{z, π X2 (z)} for z C, we deduce by (7.9) ad (7.17) that if ε 0 is small eough, the (7.18) H 1 (γ 1(C)) H 1 ( X 2 ) = O(ερ) H 1 (C). Next we prove (7.19) H 1 (Y 1 ) H 1 (Y 2 ) = O(ερ) H 1 (C). Let z C. For i = 1, 2, γ i (z) = Y cov{z, π Xi (z)} exists by (7.15), hece the relative boudary of Y i is γ i ( C). It follows by (7.16) that z γ i (z) 4ρ ε, ad the 2 discussio above shows that z γi (z) 3 z γ i (z), ξ 2. Next we defie x i = π Xi (z). Sice d(γ 1(z), X 2 ) 4ε 6 ρ by (7.17), ad there exists a ball of radius 1 2 touchig X 2 from iside at x 2, we deduce that the agle α 2 of u X2 (x 2 ) ad u X1 (γ 1(z)) is at most 12ε 3 ρ. It follows that γ 1(z) x 1 = O(ε 3 ρ) γ 1(z) z = O(ερ 3 2 ), hece the agle α1 of u X1 (x 1 ) ad u X1 (γ 1(z)) is O(ερ 3 2 ) accordig to (7.4). Therefore choosig ε 0 small eough, we have (z γ 1 (z), o, z γ 2 (z)) α 1 + α 2 = O(ε 3 ρ) < 1 8 ε2 ρ. I particular, it follows by Propositio 7.1 ad γ 1 (z) z 4ρ ε that 2 (7.20) γ 2 (z) γ 1 (z) ρ 3/2, hece (7.19) is a cosequece of (7.21) H 1[ Y ( γ 1 (relbd C) + ρ 3 2 B )] = O(ερ) H 1 (C). To prove (7.21), let τ = ρ 4ε, ad let z 1,...,z k C be a maximal family of poits with the property that z i z j 3ρ 3 2 for i j. Sice z i + ρ 3 2 B 1 are pairwise

18 20 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche disjoit for i = 1,...,k, ad each is cotaied i the differece of (1 + ρ3/2 τ )C ad (1 ρ3/2 τ )C, we deduce that ( ρ 3/2 ) (7.22) k = O H 1 (C) (ρ 3/2 ) ( 1) = O(ερ) H 1 (C) (ρ 3/2 ) ( 1). τ Now let y Y satisfy y γ 1 (z) ρ 3/2 for some z C. There exists some z i such that z i z 3ρ 3/2, hece π X1 (z i ) π X1 (z) 3ρ 3/2. I particular (7.4) implies that the agle betwee z i γ 1 (z i ) ad z γ 1 (z), which is the agle betwee u X1 (π X1 (z i )) ad u X1 (π X1 (z)), is at most 4ρ 3/2 (after choosig ε 0 small eough). Thus Propositio 7.1 yields that γ 1 (z i ) γ 1 (z) 7ρ 3/2, hece γ 1 (z i ) y 8ρ 3/2. We deduce by (7.22) that H 1[ Y ( γ 1 ( C) + ρ 3 2 B )] k H 1[ Y ( γ 1 (z i ) + 8ρ 3 2 B )] k S(8ρ 3 2 B ) = O(ερ) H 1 (C). We coclude (7.21), ad i tur (7.19). Next applyig the argumet above to X 1 as Y, γ 1 as γ 2 ad π X1 as γ 1, we deduce first the aalogue of (7.20), amely, (7.23) γ 1(z) π X1 (z) ρ 3 2, ad secodly the aalogue of (7.19), amely, (7.24) H 1 (γ 1(C)) H 1 ( X 1 ) = O(ερ) H 1 (C). Therefore combiig (7.18), (7.19) ad (7.24) yields (7.13). For (7.14), we observe that X 1 cuts Ω( X 2, Y 2 ) ito Ω = Ω( X 2, γ 1(C)) ad the closure Ω of Ω( X 2, Y 2 )\Ω. It follows by (7.8) ad (7.17) that (7.25) V (Ω ) = O(ερ) H 1 ( X 2 ) = O(ερ) H 1 (C). We deduce by (7.23) ad f 1 (y) h(y) 2ρ that [Ω( X 1, Y 1 )\Ω ] [Ω \Ω( X 1, Y 1 )] π X1 ( C) + 5ρB. Let z 1,..., z k C be a maximal system of poits i C such that z i z j 3ρ for i j. We deduce usig a argumet as above ( ρ ) k = O H 1 (C) ρ ( 1) = O(ερ 3/2 ) H 1 (C) ρ. τ Let x E satisfy x π X1 (z) 5ρ for z C. Now there exists z i C such that z i z 3ρ, hece x π X1 ( z i ) 8ρ. It follows that (7.26) V (Ω( X 1, Y 1 )) V (Ω ) k V (π X1 ( z i ) + 8ρB ) k V (8ρ B ) = O(ερ) H 1 (C). Sice V (Ω( X 2, Y 2 )) = V (Ω ) + V (Ω ), combiig (7.25) ad (7.26) completes the proof of Lemma 7.2.

19 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 21 8 Trasfer Lemma for Paraboloids for the Cases of Surface Area ad Volume We will trasfer itegrals betwee patches o paraboloids ad i E 1 usig Lemma 8.1 below. For give ω [1, 2], we cosider the paraboloid that is the graph of ϕ ω (y) = ω 2 y 2 o E 1. The derivative satisfies (8.1) ϕ ω (y) = ω y 2 y, hece if x = (y, ϕ ω (y )) ad x = (y, ϕ ω (y )) satisfy y, y tb 1 for t > 0, the (8.2) y y x x (1 + 2t 2 ) y y. Next let y 1,..., y k E 1 ad let ν 1,...,ν k 0. We observe that l i (z) = ϕ ω (y i ), z y i + ϕ ω (y i ) is the liear fuctio whose graph is the taget hyperplae to Γ(ϕ ω ) at x i = (y i, ϕ ω (y i )), ad defie ψ i (z) = l i (z) ν i. I particular for ay z E 1, the Taylor formula (see (7.2)) for ϕ ω yields (8.3) ϕ ω (z) ψ i (z) = ω 2 (z y i) 2 + ν i. Let Π 1,...,Π k be a family of pairwise o-overlappig covex polytopes i E 1, which cover a covex body C E 1 i a way such that each Π i C has o-empty iterior, ad satisfy ω 2 z y i 2 + ν i ω 2 z y j 2 + ν j for z Π i ad j = 1,...,k. We defie ψ : k Π i R by ψ(z) = ψ i (z) for z Π i, ad observe that Y = Γ(ψ) is a covex piecewise liear hypersurface. Let F i be the graph of ψ above Π i, hece F 1,...,F k are the facets of Y. We defie X = π Γ(ϕω )(C), ad assume that i = 1,...,k are the idices satisfyig that π Γ(ϕω )(F i ) itersects X i a set of positive measure for some k k. Let ν i deote the distace of x i from aff F i for i k. Lemma 8.1 We use the otatio as above. Let ε (0, ε 0 ), ad let ρ (0, ε 22 ) where ε 0 (0, 1 16 ) depeds oly o. We assume ρ 4ε B 1 C 2 ρ ε B 1 ad ω [1, 1 + ε]. Moreover, (8.4) ω 2 z y i 2 + ν i 2ρ if i = 1,...,k ad z Π i. If i additio the family V of the vertices of all Π i satisfies y z 1 8 ε ρ for y z V, the for η [0, 1] we have k X π Γ(ϕ ω )(F i ) ην i x x i 2 dx = k Π i C ην i z y i 2 dz + O(ερ) H 1 (C). Moreover, H 1 (C) = (1 + O(ε))H 1 (X), ad for z Π i ad v = (z, ψ i (z)), i = 1,...,k we have (8.5) (1 + ε) 1 d(v, Γ(ϕ ω )) < ν i z y i 2 < (1 + ε) d(v, Γ(ϕ ω )).

20 22 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche Proof We write π E 1( ) to deote the orthogoal projectio ito E 1. We observe that ξ = (o, 1) is the exterior uit vector to Γ(ϕ ω ) at the origi, ad π E 1(X) C. Let z Π i, i = 1,...,k, let v = (z, ψ i (z)), ad let x = π Γ(ϕω )(v). Combiig (8.1), (8.3) ad (8.4) yields that (u X (x), o, ξ) = O( ρ ε ) ad π E 1(x) z = O( ρ ρ ε ). Sice ρ/ε 2 < ε 2, we deduce H 1 (C) = (1 + O(ε))H 1 (X) ad (8.5) for small ε 0. Writig C i to deote the orthogoal projectio of π Γ(ϕω )(F i ) X ito E 1 for i k, it also follows that C i C, ad δ H (C i, Π i C) γ 0 ρ ρ ε where γ 0 > 0 depeds oly o. I additio (8.2) ad (8.3), ρ/ε 2 < ε ad ν i = [1 + O(ρ/ε 2 )] ν i imply that k π(f i ) X ην i x x i 2 dx = k I particular Lemma 8.1 follows from the iequalities k H 1( C\ ( k C i ην i z y i 2 dz + O(ερ) H 1 (C). C i ) ) = O(ε) H 1 (C), [ H 1 (Π i \C i ) + H 1 (C i \Π i ) ] = O(ε) H 1 (C). Sice d(x, Γ(ϕ ω )) = O(ρ/ε 2 ) for x C accordig to (8.1) ad (8.3), we deduce C\ ( k ) ρ ρ C i C + γ1 ε 3 B 1 C + γ 1 ρb 1 for some positive costat γ 1 4 depedig oly o. I additio, the diameter of ay Π i is at most 4 ρ γ 1 ρ. Therefore to prove Lemma 8.1, it is sufficiet to verify the pair of iequalities (8.6) (8.7) Π i relit C H 1( C + γ 1 ρb 1 ) = O(ε)H 1 (C), H 1( Π i ( Π i + γ 0 ρ ρ ε B 1)) = O(ε)H 1 (C). Here (8.6) readily holds by ρ 4ε B 1 C. We observe that if x relit Π i ad d(x, Π i ) = ω, the there exists a ( 2)-face F such that x + ωb 1 touches aff F i a poit of F. As ρ 4ε B 1 C, (8.7) follows from the estimate Π i relit C F Π i ( 2)-face H 2 (F) ρ ρ ε ( ρ = O(ε) ε ) 1.

21 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 23 We write S to deote the set of ( 2)-faces of ay Π i that lies i relit C, ad observe that ay F S is the ( 2)-face of exactly two Π i. Sice each F S is of diameter at most 4 ρ, we have H 2 (F) < O( ρ 2 ). Therefore writig #S to deote the cardiality of S, (8.7) follows if (8.8) ρ #S = O(ε ( 3) ). The coditio o the family V of the vertices of Π s yields that #V = O(ε 2( 1) ). We choose 1 vertices for each F S i such a way that the 1 vertices do ot lie i ay affie ( 3)-plae. Thus #S is the umber of such ( 1)-tuples, which is O(ε 2( 1)2 ). Therefore (8.8), ad i tur Lemma 8.1 are the cosequeces of ρ < ε 22. Whe comparig patches o paraboloids ad o the sphere, we eed to kow how closely paraboloids approximate the sphere. The part of S 1 below E 1 is the graph of the fuctio ϕ(y) = 1 y 2 defied o B 1, ad if y 1 2 B, the It follows that if y 1 2 B, the y 2 ϕ(y) y 2 + y 4. (8.9) 1 + ϕ 1 (y) ϕ(y) 1 + ϕ 1+2 y 2(y). I additio writig q y to deote the quadratic form represetig the secod derivative of ϕ at y, if y 1 3 B ad z E 1, the z 2 q y (z) (1 + 2 y 2 ) z 2. 9 Proof of Theorem 2.1 i the Cases of Volume ad Surface Area We assume that 4, because if 3, the Theorem 2.1 is covered [4] i the cases of surface area ad volume. The proofs of Theorem 2.1 i the cases of volume ad surface area follow the very same patter. We preset the argumet oly i the case of the surface area, because it is the more ivolved case. Accordig to Lemma 5.1, S(Q r ) S(B ) lim if = θ S () r 1 + r 1 is fiite ad positive. Therefore Theorem 2.1 i the case of the surface area follows if, for ay ε (0, ε) ad r (1, r) where ε > 0 depeds o ad r > 1 depeds o ad ε, there exists Q r,ε F r such that (9.1) S(Q r,ε ) S(B ) θ S () (r 1) + O(ε(r 1)). Here ε is at most the ε 0 s of Lemma 7.2 ad Lemma 8.1. First we defie r. Namely, it follows by the defiitio of θ S () that there exists r (1, 1 + ε 22 ) such that S(Q ñ r ) S(B ) θ S () ( r 1) + O(ε( r 1)).

22 24 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche Let r (1, r) which we fix for the rest of the proof of Theorem 2.1. We defie ow a auxiliary circumscribed polytope that will determie patches o S 1 of size r 1/ε. We choose a maximal family s1,...,s m S 1 with the property that s i s j r 1/ε for i j, ad we write G 1,...,G m to deote the facets of the circumscribed polytope whose facets touch B at s 1,...,s m. Writig B 1 j to deote the uit ( 1)-ball of cetre o cotaied i the liear ( 1)-space parallel to aff G j, we have (9.2) s j + (1 + ε) r 1 r 1 B 1 j G j s j + B 1 j. 4ε ε The Q r,ε i (9.1) will be defied as the covex hull of Γ 1,...,Γ m costructed i Propositio 9.1 (see (9.16)). Propositio 9.1 Let j = 1,...,m. Usig the otatio as above, there exists a covex piecewise liear surface Γ j satisfyig the followig properties: Γ j itersects G j ad the orthogoal projectio of Γ j ito aff G j covers G j. I additio if F is a facet of Γ j, the aff F does ot itersect it B, the orthogoal projectio of F ito aff G j itersects G j, F is a ( 1)-polytope, ad if v is a vertex of F, the (9.3) r 1 d(v, B ) 2(r 1). Moreover, if X j = π S 1(G j ) ad Y j Γ j satisfies X j = π S 1(Y j ), the H 1 (Y j ) H 1 (X j ) H 1 (X j ) S(B ) θ S () (r 1) + O(ε) (r 1) H 1 (X j ). Proof We recall that F r deotes the family of all C Fr satisfyig ext C rs 1, ad that Q r F r. Lemma 3.1 provides a polytope Q ε F r ñ such that the distace betwee ay two vertices of Q ε is at least ε r 1, ad (9.4) S( Q ε ) S(B ) θ S () ( r 1) + O(ε( r 1)). We write F 1,..., F l to deote the facets of Q ε. I additio we write w i to deote the exterior uit ormal to F i, ad defie ρ i = d( w i, aff F i ). Let f be a bouded measurable fuctio o S 1 such that f (x) = (1 + ρ i) 1 x, w i 1 for i = 1,..., l ad x π S 1(relit F i ). Sice y x = 1 + ρ i x, w i 1 for ay y F i ad x = π S 1(y), if Y Q ε is a covex hypersurface ad X = π S 1(Y ) the Lemma 6.1 yields (9.5) H 1 (Y ) H 1 (X) = f (x) dx. X

23 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 25 We defie G j = s j + λ (G j s j ) for λ = r 1 (1 + ε) r 1, ad let X j = π S 1( G j ). The Lemma 5.2 yields the existece of g SO() such that (9.6) f (x) dx H 1 ( X j ) gex j S(B ) f (x) dx. S 1 We may assume that π S 1( F i ) itersects g X j i a set of positive ( 1)-measure if ad oly if i k for k l. Let Ỹ j Q ε satisfy that π S 1(Ỹ j ) = g X j. We deduce by (9.4), (9.5), ad (9.6) that (9.7) H 1 (Ỹ j ) H 1 (g X j ) H 1 ( X j ) S(B ) θ S () ( r 1) + O(ε) ( r 1) H 1 ( G j ). We may assume that s j = ξ = (o, 1) ad g is the idetity, hece aff G j is parallel to E 1. We write C j to deote the orthogoal projectio of G j ito E 1, which satisfies (see (9.2)), r 1 4ε B 1 C j r 1 ε B 1. Let us recall that ϕ(y) = 1 y 2 ad ϕ ω (y) = ω 2 y 2 for y B 1. It follows by (8.9) ad r < 1 + ε 22 that if y 2 r 1 ε B 1, the 1 + ϕ 1 (y) ϕ(y) 1 + ϕ ω (y) for ω = 1 + ε 8. I particular the graph Γ ω of 1 + ϕ ω above 4 r 1 ε B 1 satisfies Γ ω B. Therefore we defie Z j = π e Γ ω ( G j ), ad Ỹ j Q ε by π e Γ ω (Ỹ j ) = Z j, ad we deduce by Lemma 7.2 ad (9.7) that (9.8) H 1 (Ỹ j ) H 1 ( Z j ) = H 1 (Ỹ j ) H 1 ( X j ) + O(ε( r 1)) H 1 ( G j ) H 1 ( G j ) S(B ) θ S () ( r 1) + O(ε) ( r 1) H 1 ( G j ). We may assume that i = 1,..., k are the idices satisfyig that F i itersects Ỹ j i a set of positive measure for suitable k l. For i k, let x i Z j be the poit where

24 26 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche the taget hyperplae to Z j is parallel to aff F i, ad let ν i deote the distace of x i from aff F i. We deduce by (7.10) ad (9.8) that (9.9) k πz e (ef i ) j ( 1) ν i + 2 x x i 2 dx H 1 ( G j ) S(B θ S () ( r 1) ) + O(ε)( r 1) H 1 ( G j ). Let Γ j be the uio of facets of Q ε whose orthogoal projectio ito aff G j itersects G j i a set of positive ( 1)-measure. We assume that F 1,..., F k are the facets cotaied i Γ j for suitable k, k k l. For i k, we write Π i ad ỹ i to deote the orthogoal projectio of F i ad x i, respectively, ito E 1, ad defie ν i by the property that (ỹ i, 1 + ϕ ω (ỹ i ) ν i ) aff F i. Writig C j to deote the orthogoal projectio of G j ito E 1, combiig (9.9) ad Lemma 8.1, yields that (9.10) k ( 1) ν i + ec j eπ i 2 z ỹ i 2 dx H 1 ( G j ) S(B ) θ S ()( r 1) + O(ε) ( r 1) H 1 ( G j ). I additio, if z is a vertex of Π i for i k ad v is the correspodig vertex of F i, the (9.11) (9.12) ν i z ỹ i ε d(v, Γ ω ) ε d(v, B ) = 1 ( r 1), 1 + ε ν i z ỹ i 2 (1 + ε)d(v, Γ ω ) (1 + ε) 2 ( r 1). Now we defie C j = λ 1 C j, hece G j = C j + ξ. Moreover, if i k, the let Π i = λ 1 Πi, y i = λ 1 ỹ i, ad ν i = λ 2 ν i. We coclude by (9.10) that (9.13) k ( 1)ν i + 2 z y i 2 dx H 1 (G j ) C j Π i S(B ) θ S ()(r 1) + O(ε) (r 1) H 1 (G j ). We defie ϕ(z) = 1 + z 2 for z 4 r 1 ε B 1, ad observe that Γ(ϕ) it B = accordig to (8.9). We write l i to deote the liear fuctio whose graph is the taget hyperplae to Γ(ϕ) at x i = (y i, ϕ(y i )), ad defieψ i (z) = l i (z) ν i. I additio, we defie ψ : k Π i R by ψ(z) = ψ i (z) for z Π i, ad observe that Γ j = Γ(ψ) is a covex piecewise liear hypersurface. Let F i be the graph of ψ above Π i, hece

25 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 27 F 1,...,F k are the facets of Γ j. If z is a vertex of Π i for i k ad v is the correspodig vertex of F i, the we deduce by Lemma 8.1, (9.11), ad (9.12) that (9.14) (9.15) d(v, Γ(ϕ)) (1 + ε)(ν i z y i 2 ) = λ 2 (1 + ε)( ν i λ 1 z ỹ i 2 ) (1 + ε) 5 (r 1); d(v, Γ(ϕ)) 1 1+ε (ν i z y i 2 ) = 1 1+ε λ 2 ( ν i λ 1 z ỹ i 2 ) r 1. Now combiig (9.14) ad (9.15) yields (9.3). We defie Z j = π Γ(ϕ) (G j ), assume that i = 1,...,k are the idices satisfyig that π Γ(ϕ) (F i ) itersects Z j i a set of positive measure for some k k, ad write ν i to deote the distace of x i from aff F i for i k. We recall that X j = π S 1(G j ), ad write Y j ad Y j to deote the subset of Γ j satisfyig X j = π S 1(Y j ) ad Z j = π Γ(ϕ) (Y j ), respectively. It follows first by Lemma 7.2, secodly by (7.10), ad thirdly by Lemma 8.1 ad (9.13) that H 1 (Y j ) H 1 (X j ) = H 1 (Y j ) H 1 (Z j ) + O(ε(r 1)) H 1 (G j ) k = ( 1)ν i + 2 x x i 2 dx π Γ(ϕ) (F i ) Z j + O(ε(r 1)) H 1 (G j ) H 1 (X j ) S(B ) I tur we coclude Propositio 9.1. θ S () (r 1) + O(ε) (r 1) H 1 (X j ). For the rest of the proof, we use the otatio of Propositio 9.1. We defie (9.16) Q r,ε = cov{γ 1,...,Γ m }. The Q r,ε F r ad Q r,ε (1 + 2(r 1))B. We defie W j to be the part of Q r,ε satisfyig π S 1(W j ) = X j, ad prove that for some γ > 0 depedig oly o ad idepedet of j, (9.17) H 1 (W j ) H 1 (X j ) (1 + γ ε) H 1 (X j ) S(B ) θ S ()(r 1). Let X 0 j = π S 1(s j + (1 96ε)(G j s j )), ad let Y 0 j be the part of Γ j satisfyig π S 1(Y 0 j ) = X0 j. Now if H (1 + 2(r 1))B is a compact covex set whose affie hull avoids it B, the diam H 6 r 1. Therefore if F is a facet of Q r,ε such that F itersects Γ t for some t j ad π S 1(F) X j, the π S 1(F)

26 28 K. Böröczky, K. J. Böröczky, C. Schütt, ad G. Witsche relbd X j + 12 r 1 B. Sice π S 1(x) π S 1(x ) 1 2 x x for x, x G j ad s j + r 1 4ε B 1 j G j, we deduce that Y 0 j W j ad H 1 (X j \X 0 j ) = O(ε)H 1 (X j ). Therefore (7.9) yields (9.18) H 1 (W j \Y 0 j ) H 1 (X j \X 0 j ) = O(ε) (r 1) H 1 (X j ). I additio, Propositio 9.1 implies that (9.19) H 1 (Y 0 j ) H 1 (X 0 j ) H 1 (Y j ) H 1 (X j ) (1 + O(ε)) H 1 (X j ) S(B ) θ S ()(r 1), hece combiig (9.18) ad (9.19) leads to (9.17). Addig (9.17) for j = 1,...,m proves (9.1), ad i tur Theorem 2.1 i the case of the surface area. As we stated at the begiig, the proof i the case of the volume is quite aalogous, thus we do ot preset it. 10 Trasfer Lemma i the Case of Mea Width We will trasfer itegrals betwee patches o the sphere ad i E 1 usig Lemma We recall that ξ = (o, 1) E, ad ϕ(y) = 1 y 2 parametrizes the lower hemisphere of S 1 o B 1. Lemma 10.1 Let ε (0, ε 0 ), ad let ρ (0, ε 4 ) where ε 0 is a suitable positive costat depedig oly o. I additio let C be a compact covex set satisfyig ρ 4ε B 1 C 2 ρ ε B 1, ad let y 1,..., y k E 1 such that for ay z C + 2 ρ B 1 there exists y i satisfyig 1 2 z y i 2 2ρ. Writig X = π S 1(C + ξ) ad x i = (y i, ϕ(y i )), we have H 1 (C) = (1 + O(ε))H 1 (X), d(c, X) ρ for the graph of ϕ above C, ad 1 mi{1 x, x i } dx = mi i i 2 z y i 2 dz + O(ερ) H 1 (C). X Moreover, if z C + 2 ρb 1, the x = (z, ϕ(z)) satisfies C (1 + ε) 1 1 mi i 2 z y i 2 mi{1 x, x i } (1 + ε) mi i 2 z y i 2. Proof The mai observatio is the followig fact: if z, z tb 1 for t (0, 1 2 ), the x = (z, ϕ(z)) ad x = (z, ϕ(z )) satisfies i 1 1 x, x = (1 + O(t 2 )) 1 2 z z 2, ad x π S 1(z) = O(t 3 ). Sice ρ ε < ε 0 ε, choosig ε 0 small eough, we have the followig properties: let x = (z, ϕ(z)) for z C + 2 ρ B 1, ad let mi i 1 2 z y i 2 = 1 2 z y j 2 ad mi i {1 x, x i } = 1 x, x l.

27 Covex Bodies of Miimal Volume, Surface Area ad Mea Width 29 The first, X (1 + ε) z y j 2 (1 + ε) z y l 2 1 x, x l 1 x, x j (1 + ε) 1 2 z y j 2. Secodly, writig X to deote the orthogoal projectio of X ito E 1, we have H 1 (X ) = (1 + O(ε))H 1 (X) ad d H (X, C) 1 2 ρ. Fially, 1 mi{1 x, x i } dx = mi i X i 2 z y i 2 dz + O(ερ) H 1 (X ). I tur we coclude Lemma Proof of Theorem 2.1 i the Case of Mea Width We assume that 4 because if 3, the Theorem 2.1 i the case of the mea width is covered by [3] for = 2 (as mea width is proportioal with the perimeter i this case), ad by [4] for = 3. First we preset two formulae related to the differece of the mea width of a ball ad a polytope. If P is a polytope with vertices x 1,...,x m S 1, the (11.1) M(B ) M(P) = 2 S(B ) mi(1 x, x i ) dx. S 1 i I additio if 1 r B P ad mi i (1 x, x i ) = 1 x, x l for x S 1 the (11.2) x x l r. I the case of mea width, it will be coveiet to cosider the family F r of all covex bodies that cotai 1 r B, ad whose extreme poits lie o S 1. I particular 1 r W r F r. Accordig to Lemma 5.1 ad M(rB ) M(B ) = 2(r 1), M(W r ) M(B ) lim if = θ M () r 1 + r 1 is positive ad at most two. Therefore Theorem 2.1 i the case of the mea width follows if for ay ε (0, ε 0 ) ad r > r where ε 0 depeds o ad r depeds o ad ε, there exists W r,ε F r such that (11.3) M(B ) M(W r,ε ) (2 θ M ()) (r 1) + O(ε(r 1)). Here ε 0 is at most the costat of Lemma It follows by the defiitio of θ M () that there exists r (1, 1 + ε 4 ) such that (11.4) M(B ) M(W ñ r ) (2 θ M()) ( r 1) + O(ε( r 1)). Now let r > r. We choose a maximal family s 1,...,s m S 1 with the property that s i s j r 1/ε for i j, ad we write G 1,...,G m to deote the facets of the circumscribed polytope whose facets touch B at s 1,...,s m. I additio let X j = π S 1G j.