Mean width of inscribed random polytopes in a reasonably smooth convex body

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1 Mean width of inscribed random polytopes in a reasonably smooth convex body K. J. Böröczky, F. Fodor, M. Reitzner, V. Vígh Abstract Let K be a convex body in R d and let X n = (x 1,..., x n ) be a random sample of n independent points in K chosen according to the uniform distribution. The convex hull K n of X n is a random polytope inscribed in K, and we consider its mean width W (K n ). In this article, we assume that K has with a rolling ball of radius ϱ >. First we extend the asymptotic formula for the expectation of W (K) W (K n ) earlier known in the case when K has positive Gaussian curvature. In addition, we determine the order of the variance of W (K n ), and prove the strong law of large numbers for W (K n ). We note that the strong law of large numbers for any quermassintegral was only known if K has positive Gaussian curvature. 1 Introduction and Results The convex hull of n independent, uniformly distributed random points in a given convex body K in R d is a type of random polytope that has been studied extensively (basic references are found in the surveys [3] and [8], see also [14]). As in the seminal papers of Rényi and Sulanke [, 3] (restricted to the planar case), which initiated this line of research, most of the investigations deal with asymptotic results, for n tending to infinity. We note that circumscribed polytopes have also been investigated, among others, by A. Rényi and R. Sulanke [4], F.J. Kaltenbach [18], and K.J. Böröczky and M. Reitzner [9]. Supported by OTKA grants and 4931, and by the EU Marie Curie grants Discconvgeo and PHD Supported by Hungarian OTKA grant 68398, and by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences. 1

2 We are interested in asymptotic results on the approximation orders of general convex bodies by random polytopes. We write g(n) h(n) if g(n) lim n = 1. Let K be a convex body in h(n) Rd with V (K) = 1, and let K n denote the convex hull of n independent, uniformly (according to the Lebesgue measure) distributed random points in K. By W ( ) and V ( ) we denote, respectively, mean width and volume. Upper and lower bounds for the order of magnitude of the expectation of the mean width difference were determined by R. Schneider [6]. According to Schneider s theorem there exist constants γ 1, γ > depending on K such that γ 1 n /() < W (K) EW (K n ) < γ n 1/d. (1) The upper bound in (1) is of optimal order for polytopes. This can be verified, for example, with the help of (5). Let C+ k denote the set of all convex bodies with boundary of differentiability class C k and with Gaussian curvature κ(x) > for all x K. For the case when K is C+, 3 and hence κ(x) > for all x K, R. Schneider, J.A. Wieacker [9] proved that W (K) EW (K n ) Γ( ) d(d + 1) d 1 κd κ d 1 K κ(x) d+ 1 dx, () n where κ d is the volume of the Euclidean d-dimensional unit ball. M. Reitzner [] extended the asymptotic formula () to convex bodies with C + boundary. In the case when the boundary of K is C k + for k 4, an asymptotic expansion of the expectation was obtained by P.M. Gruber [13] and M. Reitzner []. In this paper, we further extend the class of convex bodies for which () holds. We say that a convex body K has a rolling ball if there exists a ϱ > such that any x K lies in some ball of radius ϱ contained in K. According to D. Hug [17], the existence of a rolling ball is equivalent saying that the exterior unit normal at x K is a Lipschitz function of x. In particular, if K is C then K has a rolling ball, which was already observed by W. Blaschke. THEOREM 1.1 The asymptotic formula () holds for any convex body K of volume one which has a rolling ball. We note that Theorem 1.1 is close to be optimal. According to Example.1, there exists a convex body K whose boundary is C 1, and even C+ at all but one point such that lim n n (W (K) EW (Kn )) =. We recall the corresponding results about the expectation of volume for comparison. I. Bárány and D.G. Larman [7] proved that there exist constants

3 γ 1, γ > depending on K such that γ 1 n 1 (log n) d 1 < V (K) EV (K n ) < γ n /(). (3) Here, as opposed to (1), the lower bound is optimal for polytopes, and the upper bound is optimal for smooth convex bodies. On the one hand, I. Bárány and Ch. Buchta [6] provided an asymptotic formula for the case when K is a polytope. On the other hand, generalizing a result of I. Bárány [3] for convex bodies with C+ 3 boundaries, C. Schütt [3] proved that if κ(x) > for a set of x K of positive (d 1)-measure then V (K) EV (K n ) c K κ(x) 1 dx n /(), where the constant c depends only on d. Here the integral above is the so-called affine surface area. Furthermore, M. Reitzner [19] proved that the strong law of large numbers holds in the case of random volume approximation of convex bodies with C+ boundary. This result was made possible by the the upper bound on the variance of the volume of optimal order obtained in [19]. A matching lower bound on the variance is proved in I. Bárány and M. Reitzner [8] for any convex body. In this article, we prove the analogous estimates on the variance of the mean width for convex bodies with a rolling ball. We note that in the case of random approximation, upper bounds of optimal order on the variance have been proved only for convex bodies that are either polytopes or have C+ boundary (see say M. Reitzner [1], V. Vu [31] and I. Bárány and M. Reitzner [8]). THEOREM 1. If K is a d-dimensional convex body of volume one with a rolling ball then γ 1 n d+3 < VarW (Kn ) < γ n d+3, where the positive constants γ 1, γ depend on K. The upper bound in Theorem 1. yields the strong law of large numbers by standard arguments. THEOREM 1.3 If K is a d-dimensional convex body of volume one with a rolling ball then lim (W (K) W (K Γ( n))n ) = κ(x) d+ n d(d + 1) d 1 κd κ dx K d 1 with probability 1. 3

4 Some general estimates about the mean width of a random polytope We write H d 1 to denote the (d 1)-dimensional Hausdorff measure. The scalar product is denoted by,, the Euclidean unit ball in R d centred at the origin is denoted by B d, and B d is denoted by S d 1. For any convex body K in R d, integration with respect to the (d 1)- dimensional Hausdorff measure on K is denoted by dx. We say K K is twice differentiable in the generalized sense at an x K if there exists a quadratic form Q on R d 1 with the following property: If K is positioned in a way such that x = o and R d 1 is a tangent hyperplane to K from below then a neighbourhood of o on K is the graph of a convex function f on a (d 1)-dimensional ball in R d 1 satisfying f(z) = 1 Q(z) + o( z ) (4) as z tends to zero. In this case the generalized Gaussian curvature at x is κ(x) = det Q. According to the Alexandrov theorem (see R. Schneider [7] or P.M. Gruber [15]), the boundary K is twice differentiable in the generalized sense almost everywhere. For any compact convex set M in R d, we write h M to denote its support function; namely, h M (u) = max x M u, x. In particular, the width of M in the direction u S d 1 is w M (u) = h M (u) + h M ( u), and the mean width is W (M) = 1 dκ d S d 1 w M (u) du = dκ d S d 1 h M (u) du. Let K be a convex body in R d with volume one. The implied constant in O( ) in the formulas below depends on K. We start with examining the expectation of the mean width following ideas set forth in R. Schneider, J.A. Wieacker [9]. For t and u S d 1, let C(u, t) = {x K : u, x h K (u) t}. For x 1,..., x n K, we usually write X n = (x 1,..., x n ) and K n = [x 1,..., x n ], and we define the function { 1 if t < hk (u) h ϕ(t, u, X n ) = Kn (u) otherwise. In particular, for fixed t and u, ϕ(t, u, X n ) = 1 if and only if none of x 1,..., x n 4

5 lie in C(u, t). We deduce, using the Fubini theorem, that E(W (K) W (K n )) = h K (u) h Kn (u) du dx n dκ d K n S d 1 = wk (u) ϕ(t, u, X n ) dt du dx n dκ d K n S d 1 = wk (u) (1 V (C(u, t))) n dt du. dκ d S d 1 There exist γ, n > depending on K such that V (C(u, t)) > 3 ln n n n > n, u S d 1 and t > γ ( ln n) 1 d. Therefore, if n > n n then E(W (K) W (K n )) = dκ d S d 1 γ ( ln n n ) d 1 for any (1 V (C(u, t))) n dt du + O(n 3 ). Example.1 If K is a convex body in R d such that o K, K is C+ on K\o, and the graph of f(x) = x 3 3d on R d 1 B d is part of K then E(W (K) W (K n )) γ n 4d 3d +1 where γ > depends on d and 4d <. 3d +1 Proof: We write u to denote opposite of the dth basis vector, and γ 1, γ,... to denote positive constants depending on d. As f(x) = x 1+α for α = 1, 3d simple calculations show that at all x f(x)u for x R d 1 (B d \o), the exterior unit normal u at x f(x)u to K satisfies γ 1 x α u u γ x α, and each the principal curvature is at least γ 3 x 1+α γ 4 u u 1+α α. Let Ξ(n) = S d 1 (u + n α d+α B d ). In particular if n is large, u Ξ(n) and t n 1+α d+α then (5) V (C(u, t)) γ 5 t n 1 α d+α d 1 γ 5 n 1. Therefore (5) yields E(W (K) W (K n )) γ 6 Ξ(n) n 1+α d+α du γ7 n dα+1 d+α. Next, we estimate the variance. According to the Efron-Stein jackknife inequality (see M. Reitzner [19]), we have that VarW (K n ) (n + 1)E(W (K n+1 ) W (K n )). (6) We write f g if f γg for a constant γ > depending only on K. For t, u S d 1 and x 1,..., x n+1 K, let X n+1 = (x 1,..., x n+1 ), K n+1 = [x 1,..., x n+1 ] and K n = [x 1,..., x n ]. Further, we define the function { 1 if hkn (u) t h ϕ(t, u, X n+1 ) = Kn+1(u) otherwise. 5

6 We set the volume of the empty set to be zero. It follows by the Efron-Stein jackknife inequality and the Fubini theorem that VarW (K n ) ( ) n h Kn+1 (u) h Kn (u) du dx n+1 K n+1 S d 1 = n K n+1 S d 1 (h Kn+1 (u) h Kn (u)) S d 1 (h Kn+1 (v) h Kn (v)) dv du dx n+1 wk (v) wk (u) = n ϕ(t, u, X n+1 ) K n+1 S d 1 S d 1 ϕ(s, v, X n+1 ) ds dt dv du dx n+1 wk (v) wk (u) = n V (C(u, t) C(v, s)) S d 1 S d 1 For any u S d 1 and s, t, let and for v Σ(u, t; s), let (1 V (C(u, t) C(v, s))) n ds dt dv du. Σ(u, t; s) = {v S d 1 : C(u, t) C(v, s) }, V + (u, t; v, s) = max{v (C(u, t), V (C(v, s)). Therefore our estimate of the variance yields that if n > n then VarW (K n ) n S d 1 γ ( ln n n ) d 1 3 Proof of Theorem 1.1 t Σ(u,t;s) V + (u, t; v, s) (1 V + (u, t; v, s)) n dv ds dt du + O(n ). (7) Let K be a convex body in R d with a rolling ball of radius ϱ >. We write u x to denote the exterior unit normal at x K. In particular, if f is measurable on S d 1 then by formula (.5.3) in [7] f(u) du = f(u x )κ(x) dx. (8) S d 1 K Let x K. The existence of the rolling ball yields that V (C(u x, t)) κ d 1ϱ d 1 t d + 1 for t [, ϱ]. (9) 6

7 In addition, if κ(x) exists and positive then we deduce by (4) that lim t V (C(ux, t)) = κ d 1. (1) t (d + 1)κ(x) 1 We will need some asymptotic formula using the gamma function (see E. Artin [1]). First we note that for α >, the representation of the beta function by the gamma function and the Stirling formula imply 1 lim n nα (α+1) ln n n τ α 1 (1 τ) n α Γ(α)Γ(n + 1) dτ = lim n n Γ(α + n + 1) = Γ(α). Now if τ < 1, then (1 τ) n < e nτ n (α+1). Therefore, if f(n) (, 1) satisfies f(n) for large n, then f(n) (α+1) ln n n τ α 1 (1 τ) n dτ Γ(α)n α as n tends to infinity. For β and ω >, it follows using the substitution τ = ωt that g(n) ( ) t β (1 ωt ) n (β + 1) dt Γ n (β+1) (d + 1)ω (β+1), (11) d + 1 assuming that g(n) (, ω ) for all n, and g(n) ( (α+1) ln n ω n ) for large n, where α = (β+1). Proof of Theorem 1.1. For the n coming from (5), we define θ n (u) = n dκ d γ ( ln n n ) d 1 (1 V (C(u, t))) n dt for n > n and u S d 1. According to (5), we have lim n E(W (K) W (Kn )) = lim θ n (u x )κ(x) dx. (1) n n K Since for large n, θ n (u) < γ for some γ depending only on K by (9) and (11) (with β = ) for any u S d 1, and κ(x) ϱ (d 1) for any x K, we may apply the Lebesgue dominated convergence theorem. Let x K such that κ(x) exists and positive. Now for any ε (, 1), (1) yields that there exists a t ε > such that κ d 1 (1 ε) (d + 1)κ(x) 1 t V (C(u x, t)) (1 + ε) 7 κ d 1 (d + 1)κ(x) 1 t

8 for t (, t ε ). Therefore (11) (with β = ) implies lim θ n(u x ) = n 1 κ(x) Γ( ) (d + 1) d 1 dκd κ d 1. In turn, we conclude Theorem 1.1 by (1). 4 Proof of the upper bound in Theorem 1. To prove Theorem 1., we observe that if a (, 1) then (1 a (1 )n (1 a) > + a ) n an > n, which in turn yields a(1 a) n < n ( 1 a ) n. (13) Since our estimate on the variance depends on (7), we estimate the size of Σ(u, t; s) for u S d 1. The existence of the rolling ball of radius ϱ at x C(u, t) K shows that u x u t ϱ for t ϱ. In particular, let < s t ϱ. If v Σ(u, t; s) then v u < 4ϱ 1 t 1, and hence the (d 1)-measure of Σ(u, t; s) is at most γt d 1 for some γ > depending on d. We set γ = κ d 1ϱ d 1, and simplify (7) by applying first (9) and (13), and secondly the formula (11) to obtain VarW (K n ) n S d 1 γ ( ln n n ) d 1 γ ( ln n n ) d 1 t ( t t d 1 n 1 γ t ( 1 γ t ) n ds dt du + O(n ) ) n dt + O(n ) n d+3. 5 Proof of the lower bound in Theorem 1. The idea of the proof is similar to the one in M. Reitzner ([1]); namely, VarV 1 (K n ) is at least the sum of the variances inside independent caps. First we separate the part of K where reasonably sized caps are contained in touching balls of fixed radius. Next we verify the technical estimates (15) and (18), which lead to the estimates () and (1) ensuring the independence 8

9 of the caps in the final argument. In addition we need Lemma 5.1 to estimate the variance inside a cap. For any polytope P and vertex z of P, we write N P (z) to denote the exterior normal cone to z. We recall the Alexandrov theorem (see R. Schneider [7] or P.M. Gruber [15]) that the boundary K is twice differentiable in the generalized sense almost everywhere with respect to H d 1. We deduce by (8) that the (d 1)-measure of the points x K with κ(x) > is positive. Therefore there exists R > and a Ξ K with H d 1 (Ξ ) > such that each principal curvature at all x Ξ is at least. For any x R Ξ there ϱ exists a maximal σ x (, ] such that C(u 8d x, σ x ) x Ru x + R B d, and σ x being lower semi continuous, is a measurable function of x Ξ. Therefore ϱ there exists σ (, ] such that if Ξ denotes the family of x K such 8d that C(u x, σ) x Ru x + R B d then H d 1 (Ξ) >. (14) For u S d 1 and t >, we define H(u, t) = {z : z, u = h K (u) t}. Let x Ξ, and let t (, σ). The existence of the rolling ball and the definition of Ξ imply (x tu x + ϱtb d ) H(u x, t) H(u x, t) K x tu x + RtB d. (15) Let w 1,..., w d be the vertices of a regular (d 1)-simplex in H(u x, t) whose circumcentre is x tu x, and whose circumradius is ϱt, and hence ( x tu x + ) ϱt d Bd H(u x, t) [w 1,..., w d ] K. In addition we set w = x, and for j =,..., d, j (x, t) = w j + 1 4d ([w,..., w d ] w j ). In particular (15) yields V ( j (x, t)) t for j =,..., d. (16) If z j j (x, t), j =,..., d, and v u x then ϱt/(d) h[z1,...,z d ](v) z, v ϱt t/ z, u x h [z1,...,z d ](u x ) t, (17) and hence the tangent function of the angle of u x and any u N [z,...,z d ](z ) is between t 4 and d t ϱ ϱ. Therefore defining ( ) t Σ 1 (x, t) = S d 1 u x + 8 ϱ Bd, ( Σ (x, t) = S d 1 u x + d ) t B d, ϱ 9

10 we have Σ 1 (x, t) S d 1 N [z,...,z d ](z ) Σ (x, t). (18) For j = 1,, we consider the dual cones which satisfy Σ j(x, t) = {y R d : y, v for all v Σ j (x, t)}, Σ (x, t) {t(y z ) : t and y [z, z 1,..., z d ]} Σ 1(x, t). (19) Let γ = 9 d R/ϱ, σ = σ/γ and γ = Rγ. If x Ξ and t (, σ ) then it follows by (15) that C(x, γt) x + γ tb d. () Next if z (x, t) and y H(u x, γt) K then (15) yields that the tangent of the angle of u x and y z is at most Rγt = ϱ (γ/)t 4d, therefore y z t Σ (x, t). In particular K\C(x, γt) z + Σ (x, t). (1) If A is an event in some probability space then we write I(A) to denote the indicator function. In addition for x Ξ, t (, σ ), and z i i (x, t), i =,..., d, writing F = [z 1,..., z d ] we define W F (z ) = dκ d Σ (x,t) h [z,f ](u) du. Naturally W F (z ) depends on x and t, as well, but it will be always clear what x and t are. LEMMA 5.1 If Z is a random variable chosen uniformly from (x, t) for x Ξ and t (, σ ), and z i i (x, t) for i = 1,..., d; then VarW [z1,...,z d ](Z) t. Proof: We define F = [z 1,..., z d ]. Let w be the centroid of the facet of (x, t) opposite to x, let w 1 = x + 1 w and w 3 3 = 1 x + w. In addition 3 3 we define (compare (19)) Ψ 1 = (w 1 Σ (x, t)) (x, t), Ψ = (w + Σ (x, t)) (x, t). 1

11 In particular there exists some γ > depending on K such that V (Ψ j ) γ V ( (x, t)). () Moreover for any Z 1 Ψ 1 and Z Ψ, if v S d 1 then h [Z1,F ](v) h [Z,F ](v) by [Z, z 1,..., z d ] [Z 1, z 1,..., z d ], and if even v Σ 1 (x, t) (compare (18)) then h [Z1,F ](v) h [Z,F ](v) = v, Z 1 v, Z v, w 1 v, w t. Therefore if Z 1 Ψ 1 and Z Ψ then W F (Z 1 ) W F (Z ) t H d 1 (Σ 1 (x, t)) t. In turn we deduce (compare ()) VarW F (Z) = 1 E(W F (Z 1 ) W F (Z )) 1 E[(W F (Z 1 ) W F (Z )) I(Z 1 Ψ 1, Z Ψ )] t E[I(Z 1 Ψ 1, Z Ψ )] t. It is sufficient to prove the lower bound in Theorem 1. for large enough n. We fix t n = n, (3) and hence V (C(u x, t n )) 1/n for all x Ξ. We choose a maximal family of points y 1,..., y m Ξ such that for i j, we have (compare ()) In particular (14) yields y i y j γ t n. m n d 1. (4) For j = 1,..., m, let A j denote the event that each i (y j, t n ), i =,..., d contains exactly one random point out of x 1,..., x n, and C(y j, γt n ) contains no other random point (compare (1)). We note that there exist positive α, β depending only on K such that for i =,..., d, we have V ( i (y j, t n )) α/n and V (C(y j, γt n )) β/n. 11

12 Thus for j = 1,..., m, we have ( ) n (α ) ( P{A j } 1 β n d 1 1. (5) d + 1 n n) If A j holds then we write Z j to denote the random point in (y j, t n ), and F j to denote the convex hull of the random points in i (y j, t n ) for i = 1,..., d. Hence for any u Σ (y j, t n ), (1) yields h Kn (u) = h [Zj,F j ](u) (6) given A j. In particular if 1 i < j m and A i, A j hold, then W Fi (Z i ) and W Fj (Z j ) are independent according to (18). We next introduce the sigma algebra F that keeps track of everything except the location of Z j (y j, t n ) for which A j occurs. We decompose the variance by conditioning on F: VarW (K n ) = E Var(W (K n ) F) + Var E(W (K n ) F) E(VarW (K n ) F). The independence structure mentioned above implies that Var(W (K n ) F) = Var Zj W (K n ) I(A j )=1 = I(A j )=1 Var Zj W Fj (Z j ) where the variance is taken with respect to the random variable Z j (y j, t n ), and we sum over all j = 1,..., m with I(A j ) = 1. Combining this with Lemma 5.1, (3), (4) and with (5) implies VarW (K n ) E ( m ) n E I(A j ) I(A j )=1 t n n m n d+3 6 Proof of Theorem 1.3 j=1 First, we deduce by Chebyshev s inequality that ) P ( W (K) W (K n ) E(W (K) W (K n )) n ε ε n 4 VarW (Kn ) n d 1. 1

13 is finite for n k = k 4, the sum of the probabili- ) ε ties Since the sum k= ( P n d 1 k W (K) W (K nk ) E(W (K) W (K nk )) n k for k is finite as well. Therefore the Borel-Cantelli lemma and Theorem 1.1 yield that lim (W (K) W (K n k ))n k k = Γ( ) d(d + 1) d 1 κd κ d 1 with probability 1. Now, W (K) W (K n ) is decreasing, and hence K κ(x) d+ dx (7) (W (K) W (K nk 1 ))n k 1 (W (K) W (K n))n (W (K) W (Knk ))n k n hold for n k 1 n n k. As lim k k n k 1 = 1, the subsequence limit (7) yields Theorem 1.3. Acknowledgement: We would like to acknowledge the stimulating conversations with Imre Bárány. References [1] E. Artin. The gamma function. Holt, Rinehart and Winston, [] I. Bárány: Intrinsic volumes and f-vectors of random polytopes, Math. Ann., 85 (1989), [3] I. Bárány: Random polytopes in smooth convex bodies. Mathematika 39 (199), no. 1, [4] I. Bárány: The technique of M-regions and cap-covering: a survey. Rend. Circ. Mat. Palermo () Suppl., 65 (1999), [5] I. Bárány: Random polytopes, convex bodies, and approximation. In: A. Baddeley, I. Bárány, R. Schneider, W. Weil, Stochastic Geometry (C.I.M.E. Course, Martina Franca, 4), Lecture Notes Math., Springer (to appear). [6] I. Bárány, Ch. Buchta: Random polytopes in a convex polytope, independence of shape, and concentration of vertices. Math. Ann., 97 (1993),

14 [7] I. Bárány, D.G. Larman: Convex bodies, economic cap coverings, random polytopes. Mathematika, 35 (1988), [8] I. Bárány, M. Reitzner: The central limit theorem for random polytopes in a polytope, manuscript [9] K. Böröczky Jr., M. Reitzner: Approximation of smooth convex bodies by random circumscribed polytopes. Ann. Applied Probab., 14 (4), [1] B. Efron: The convex hull of random set of points. Biometrika, 5 (1965), [11] H. Groemer: On the mean value of the volume of a random polytope in a convex set. Arch. Math., 5 (1974), [1] S. Glasauer, P.M. Gruber: Asymptotic estimates for best and stepwise approximation of convex bodies III. Forum Math., 9 (1997), [13] P.M. Gruber: Expectation of random polytopes. Manuscripta Math., 91 (1996), [14] P.M. Gruber: Comparisons of best and random approximation of convex bodies by polytopes. Rend. Circ. Mat. Palermo () Suppl., 5 (1997), [15] P.M. Gruber: Convex and discrete geometry. Springer, 7. [16] M. Hartzoulaki, G. Paouris: Quermassintegrals of a random polytope in a convex body. Arch. Math., 8 (3), [17] D. Hug: Measures, curvatures and currents in convex geometry. Habilitationsschrift, Univ. Freiburg,. [18] F.J. Kaltenbach: Asymptotisches Verhalten zufälliger konvexer Polyeder. Doctoral Thesis, Freiburg 199. [19] M. Reitzner: Random polytopes and the Efron-Stein jackknife inequality. Ann. Probab., 31 (3), [] M. Reitzner: Stochastic approximation of smooth convex bodies. Mathematika, 51 (4), [1] M. Reitzner: Central limit theorems for random polytopes. Probab. Theory Relat. Fields, 133 (5),

15 [] A. Rényi, R. Sulanke: Über die konvexe Hülle von n zufällig gewählten Punkten, Z. Wahrscheinlichkeitsth. verw. Geb., (1963), [3] A. Rényi, R. Sulanke: Über die konvexe Hülle von n zufällig gewählten Punkten, II, Z. Wahrscheinlichkeitsth. verw. Geb., 3 (1964), [4] A. Rényi, R. Sulanke: Zufällige konvexe Polygone in einem Ringgebiet. Z. Wahrscheinlichkeitsth. verw. Geb., 9 (1968), [5] Y. Rinott: On normal aproximation rates for certain sums of dependent randomvariables. J. Comput. Appl. Math., 55 (1994), , [6] R. Schneider: Approximation of convex bodies by random polytopes, Aequationes Math., 3 (1987), [7] R. Schneider: Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press, Cambridge [8] R. Schneider: Discrete aspects of Stochastic Geometry, In: J.E. Goodman, J. O Rourke (eds.), Handbook of Discrete and Computational Geometry, nd ed., CRC Press, Boca Raton 4, pp [9] R. Schneider, J.A. Wieacker: Random polytopes in a convex body. Z. Wahrsch. Verw. Gebiete, 5 (198), [3] C. Schütt: Random polytopes and affine surface area. Math. Nachr., 17 (1994), [31] V. Vu: Central limit theorems for random polytopes in a smooth convex set. Adv. Math., 7 (6), [3] W. Weil, J.A. Wieacker: Stochastic geometry. In: P.M. Gruber, J.M. Wills (eds.), Handbook of Convex Geometry, North-Holland, Amsterdam 1993, pp [33] H. Ziezold: Über die Eckenanzahl zufälliger konvexer Polygone. Izv. Akad. Nauk Armjansk. SSR, 5 (197), Károly J. Böröczky Alfréd Rényi Institute of Mathematics, PO Box 17, H-1364 Budapest, Hungary, carlos@renyi.hu and Department of Geometry, Roland Eötvös University, 15

16 Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary Ferenc Fodor Department of Geometry, University of Szeged, Aradi vértanúk tere 1, H-67 Szeged, Hungary Matthias Reitzner Institut für Diskrete Mathematik und Geometrie, TU Wien Wiedner Hauptstrasse 8-1, 14 Wien, Austria Viktor Vígh Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-67 Szeged, Hungary 16