Asymptotic mean values of Gaussian polytopes

Transcription

1 Asyptotic ean values of Gaussian polytopes Daniel Hug, Götz Olaf Munsonius and Matthias Reitzner Abstract We consider geoetric functionals of the convex hull of norally distributed rando points in Euclidean space R d. In particular, we deterine the asyptotic behaviour of the expected value of such functionals and of related geoetric probabilities, as the nuber of points increases. Introduction Let X, X,... be independent identically distributed rando points in Euclidean space R d. Geoetric functionals such as volue, surface area, ean width, or the nuber of k-faces, of the convex hull of such rando points have been studied repeatedly in the literature. A recent survey is provided in []. If the rando points are chosen fro a given copact convex set K R d with non-epty interior, it is natural to consider the unifor distribution on K. More generally, the distribution function ay have a density with respect to Lebesgue easure. In case the doain is R d, the noral distribution is a canonical choice. Another ethod of generating n + rando points in R d goes back to a suggestion by Goodan and Pollack. Let R denote a rando rotation of R n, i.e. a stochastic choice fro the orthogonal group On under noralized Haar easure, let Π d : R n R d be the projection to the first d coponents d < n, put Π := Π d R, and let v,..., v n+ be the vertices of a regular siplex T n in R n. Then Πv,..., Πv n+ are n+ rando points in R d in the Goodan-Pollack odel. Clearly, as long as one considers rotation invariant functionals of such rando points, one can project to a rando linear subspace, instead of first rotating randoly and then projecting to a fixed subspace. For further inforation on this Grassann approach and related work of Vershik and Sporyshev [5], we refer to [3]. In connection with the Goodan-Pollack odel, Affentranger and Schneider [3] especially found an expression for the expected value Ef k ΠT n of the nuber of k-faces of the rando polytope ΠT n, for 0 k < d < n, in ters of external and internal angles of T n and its faces. In addition, they showed that asyptotically Ef k ΠT n d d βt k, T d π log n d. d k + as n, where βt k, T d is the internal angle of a regular d -siplex at one of its k- diensional faces. It was also observed by these authors that the value Ef d ΠT n coincides with the expected nuber of facets of the convex hull of n + independent and norally distributed rando points in R d. An explanation for this relationship was subsequently found by Baryshnikov and Vitale [4]. To describe an iportant consequence of their result, and for later use, we call an i.i.d. sequence of standard Gaussian rando points in R d a Gaussian saple in R d. Let X,..., X n+ be a Gaussian AMS 99 subject classifications. Priary 5A, 60D05; secondary 5B, 6H0. Key words and phrases. Rando points, convex hull, f-vector, geoetric probability, noral distribution, Gaussian saple, Stochastic Geoetry.

2 saple in R d. Its convex hull will be called a Gaussian polytope in R d and denoted by [X,..., X n+ ]. Let ϕ be an affine invariant easurable functional on the convex polytopes. Then it is shown in [4] that ϕπt n d = ϕ[x,..., X n+ ],. where = d eans equality in distribution. Thus, if X,..., X n is a standard Gaussian saple and f k denotes the nuber of k-faces, cobining. and., we get Ef k [X,..., X n ] d d βt k, T d π log n d.3 d k + as n see [4, Theore 3]. A direct derivation of this asyptotic expansion has been given by Raynaud [] in the special case when k = d. A ain objective of the present work is to provide a direct derivation of.3 for all k {0,..., d }. Incidentally, the present approach leads to a new expression for the internal angles of a regular siplex. A basic idea of the geoetric part of our ethod is to characterize a k-face of a polytope by considering the projection of the vertices of the polytope to the orthogonal copleent of that face. Another geoetric tool, which we will apply repeatedly, is the classical affine Blaschke- Petkantschin forula. Thus, exploiting the fact that the projection to a subspace of a norally distributed point is again norally distributed, we can rewrite the expected value in ters of geoetric probabilities of the for PY / [Y,..., Y n k ],.4 where Y, Y,..., Y n k are independent norally distributed rando points in R d k with different variances. Note that.4 is the probability that a norally distributed rando point is contained in a Gaussian polytope in R d k. In a second step, we then derive the asyptotic behaviour of such geoetric probabilities. A ajor advantage of the present ore direct treatent of norally distributed rando points is that it can be applied to functionals which are not necessarily affine invariant. More explicitly, we are able to obtain results for a class of rotation invariant functionals that has been introduced by Wieacker [7], and has further been studied by Affentranger and Wieacker [] and Affentranger []. Particular cases of such functionals are the total k-diensional volue V k skel k P of the k-faces of a polytope P, and the nuber of k-faces f k P. For these we obtain as a consequence of a ore general result: Theore.. Let X,..., X n be i.i.d. rando points in R d with coon standard noral distribution. Then EV k skel k [X,..., X n ] c k,d log n d.5 and Ef k [X,..., X n ] c k,d log n d.6 as n, where c k,d and c k,d are constants depending only on k and d. The constants c k,d and c k,d are given in Section 4. These results copleent asyptotic expansions for the ean values of querassintegrals of Gaussian polytopes, which were given by Affentranger []. Iportant further contributions to convex hulls of norally distributed rando points are due to Hueter [7], who proved a Central Liit Theore for V d [X,..., X n ] and f 0 [X,..., X n ]. We also investigate functionals of the centrally syetric convex hull [±X,..., ±X n ], where again X,..., X n is a standard Gaussian saple in R d. It follows fro [4] that the syetric convex

3 hull of a Gaussian saple can be obtained by randoly rotating and projecting to R d a regular crosspolytope in R n. This fact was used by Böröczky and Henk [5], who thus found the surprising result that the asyptotic expansion does not change if T n in. is replaced by a regular n-diensional crosspolytope. Apart fro aditting the treatent of ore general functionals also in the syetric situation, our ethod leads to an alternative and ore direct explanation for this phenoenon. Auxiliary results In this section, we will fix our notation and provide soe auxiliary results. We will work in Euclidean spaces R n of varying diensions n. The nor in these spaces will always be denoted by. For points x,..., x R n, the convex hull of these points is denoted by [x,..., x ]. If P R n is a convex polytope, then we write F k P for the set of its k-diensional faces and f k P for the nuber of these k-faces, where k {0,..., n}. The k-diensional volue of the convex hull of k + points x 0,..., x k is denoted by k x 0,..., x k. Finally, k-diensional Lebesgue easure in a k-diensional flat E R n is denoted by λ E, or siply by λ k, if the affine subspace E is clear fro the context. The affine Blaschke-Petkantschin forula will be an iportant tool in our analysis. Let Ek n be the space of k-flats in R n, and let L n k be the space of k-diensional linear subspaces of Rn, k {0,..., n}. Both spaces are endowed with the usual topologies. The rotation invariant Haar probability easure on L n k is denoted by ν k the diension n will always be clear fro the context. Moreover, a otion invariant Haar easure on Ek n is defined by µ k := L n k L {L + y } λ L dy ν k dl, where L is the orthogonal copleent of L L n k in Rn. Then, for n, q {0,..., n} and any non-negative easurable function f : R n q+ R, the affine Blaschke-Petkantschin forula see [3, 6.] states that... fx 0,..., x q λ n dx 0... λ n dx q. R n R n = c nq q! n q... fx 0,..., x q q x 0,..., x q n q λ E dx 0... λ E dx q µ q de, where E n q E E c nq := ω n q+ ω n ω ω q and ω r := π r /Γ r, r > 0; for r N, ωr is the volue of the r -diensional unit sphere. In addition to the Blaschke-Petkantschin forula, we will require ore specific preparations related to the ultidiensional noral distribution. As usual, we fix an underlying probability space Ω, A, P. A rando point X in R n, defined on Ω, is said to be norally distributed with positive definite n n- covariance atrix Σ and ean 0 if XP has the density f Σ x = π n det Σ exp xt Σ x, x R n ; then we write X d = N0, Σ. For siplicity, we will exclusively consider the case Σ = σ I n, σ > 0. 3

4 The distribution function of the one-diensional noral distribution N0, is given by φz := π z e t dt, z R. The Landau sybols o and O, which will be used several ties in the following, are defined as usual. Moreover, writing fx gx for real-valued functions f, g, defined on a suitable subset of R, we ean that fx/gx as x. The natural logarith will be denoted by log. Finally, all constants which are used subsequently, depend only on the paraeters that are indicated. Lea.. For α, β > 0 and s R, and as β. φz β α z s exp αz dz = Γα α π α β α log β α+s + o φz β α z s exp αz dz = Γα π α β α log β α+s + o Proof. A coplete proof can be given, for instance, by refining and extending an arguent of Affentranger see [, Appendix II] or by generalizing an alternative approach indicated in [8]. The asyptotic expansion provided in Lea. will be used several ties. A first application is given in the proof of the next result, which will be needed in Section 4. There the following expressions arise naturally. For a 0, p, q, r R with p > q > r > 0 and γ R, we define I a p, q, r; γ := This quantity will be copared with as p. φz p q zs a+q r γ + z a/ exp rs γ + z q rz ds dz. J a p, q, r; γ := r φz p q z a exp qz rγ dz Lea.. Let a 0, and let p, q, r R satisfy q > r > 0. Then, uniforly in γ R, I a p, q, r; γ J a p, q, r; γ = O p q log p q+a 3 as p. Proof. By Fubini s theore, I a p, q, r; γ = φz p q z γ + z a exp q rz s a+q r exp r γ + z s ds dz.. 4

5 Repeated partial integration yields that s a+q r exp rγ + z s ds rγ + z exp rγ + z c a, q, r γ + z exp rγ + z,.3 where c a, q, r is a constant. Hence,. and.3 iply that I ap, q, r; γ φz p q z γ r + z a exp qz rγ dz c a, q, r φz p q z γ + z a 4 exp qz rγ dz. Then, for z, r > 0, a 0 and γ R, we use the estiates z γ + z a z a exp rγ c a, rz a and with a constant c a, r, to infer that z γ + z a 4 exp rγ c a, rz a 3, I a p, q, r; γ J a p, q, r; γ c 3 a, q, r φz p q z a exp qz dz, where c 3 a, q, r is a constant. Now an application of Lea. copletes the proof. We reark that a siilar result holds, with essentially the sae proof, if in the definition of I a and J a the function φ is replaced by φ. 3 Transition to probabilities Throughout this paper, X,..., X n will be independent rando points with X i d = N 0, I d. For n d +, k {0,..., d } and I {,..., n} with I = k +, we define h I x,..., x n := {[x i : i I] F k [x,..., x n ]}, x,..., x n R d, and put I 0 := {,..., k + }. By syetry, we then obtain for the ean nuber of k-faces of the P-alost surely siplicial Gaussian polytope [X,..., X n ] that Ef k [X,..., X n ] = h I X,..., X n dp = I =k+ n k + h I0 X,..., X n dp. 3. 5

6 In order to transfor this ean value into a basic geoetric probability, we define for d, k N and q 0 the constant k Md, k, q := π d k+... k x 0,..., x k q exp x i λ d dx 0... λ d dx k. R d R d i=0 In the case when q N, Md, k, q is the q-th oent of the rando k-diensional volue of a rando k-siplex [X 0,..., X k ] with k + independent and norally distributed vertices X i = N 0, d I d, i = 0,..., k. The following lea will be applied in the special case when d = k. Lea 3.. For d, k N, k d and q 0, Md, k, q = π k q c dk c q+dk q k +. k! Proof. For q N 0 this can be shown as in the proof of Satz 6.3. in [3]. The general case follows by using the connection with the Wishart distribution cf. [0], [9, p. 437, 4.5.3] and [6, pp. 303, 35]; see also [4]. Theore 3.. Let X,..., X n be n d + independent rando points in R d d with X i = N 0, I d. Then, for k {0,..., d }, n Ef k [X,..., X n ] = PY / [Y,..., Y n k ], k + where Y, Y,..., Y n k are independent rando points in R d k with Y = d N 0, k+ I d k and Y i d = N 0, I d k for i =,..., n k. Proof. Using 3., the Blaschke-Petkantschin forula. and the definition of µ k, we obtain where Ef k [X,..., X n ] n n = π d n... h I0 x,..., x n exp x i λ d dx... λ d dx n k + R d R d i= = c 4 n, k, d h I0 z + y,..., z k+ + y, x k+,..., x n R d R d L d L k L L k+ n k z,..., z k+ d k exp z i k + y x i i= i=k+ λ L dz... λ L dz k+ λ L dyν k dlλ d dx k+... λ d dx n, c 4 n, k, d := n π d n c dk k! d k. k + Assue that z,..., z k+ L are affinely independent, let z k+,..., z n L and y L. Then, for λ L -alost all y k+,..., y n L, [z + y,..., z k+ + y] F k [z + y,..., z k+ + y, z k+ + y k+,..., z n + y n ] 6

7 if and only if Hence, defining for y, y k+,..., y n R d, we obtain Ef k [X,..., X n ] [ = c 5 n, k, d... L d k L y / [y k+,..., y n ]. gy, y k+,..., y n := {y / [y k+,..., y n ]}, L ] k+ k z,..., z k+ d k exp z i λ L dz... λ L dz k+... gy, y k+,..., y n exp n y i k + y L L i=k+ λ L dy k+... λ L dy n λ L dyν k dl, i= where c 5 n, k, d := c 4 n, k, dπ kn k. Thus, by Lea 3. and the rotation invariance of the integrand, it follows that Ef k [X,..., X n ] = c 5 n, k, dπ k k+ Mk, k, d k... gy, y k+,..., y n exp R d k R d k λ d k dy k+... λ d k dy n λ d k dy. n i=k+ y i k + y Applying Lea 3. and siplifying the constants, we obtain the assertion of the theore. In the reainder of this section, we will explain how the preceding arguent can be odified to yield a siilar relation in the centrally syetric case. Moreover, the approach will be extended to cover ore general functionals. 3. The centrally syetric case Let X,..., X n be n d independent rando points in R d with X i d = N 0, I d. We write [x,..., x n ] c := [x, x,..., x n, x n ] for the centrally syetric convex hull of x,..., x n I + J = k +, we put R d. For subsets I, J {,..., n} with h IJ x,..., x n := {[x i, x j : i I, j J] F k [x,..., x n ] c } 7

8 and set I 0 := {,..., k + }, J 0 :=. Since [X,..., X n ] c is P-alost surely a siplicial polytope, by syetry and by the reflection invariance of the noral distribution, we find that Ef k [X,..., X n ] c = = k+ r=0 I =r J =k+ r k+ r=0 n r = k+ n k + h IJ X,..., X n dp n r h I0 J k + r 0 X,..., X n dp h I0 J 0 X,..., X n dp. The integral thus obtained can be further siplified as shown in the next theore. Theore 3.3. Let X,..., X n be n d independent rando points in R d d with X i = N 0, I d. Then, for k {0,..., d }, n Ef k [X,..., X n ] c = k+ PY / [Y,..., Y n k ] c, k + where Y, Y,..., Y n k are independent rando points in R d k with Y = d N 0, k+ I d k and Y i d = N 0, I d k for i =,..., n k. Proof. Repeat the proof of Theore 3. with g replaced by for y, y k+,..., y n R d. g c y, y k+,..., y n := {y / [y k+,..., y n ] c } 3. A general functional A class of functionals, which has first been introduced by Wieacker [7] and has further been studied in [], [], depends on two paraeters. For a polytope P R d, real nubers a, b 0 and k {0,..., d }, we define T d,k a,b P := ηf a λ k F b, F F k P where λ k F denotes the k-diensional Lebesgue easure of a k-diensional face F F k P calculated in the affine hull afff of F, and ηf := distafff, 0 is defined as the distance of afff fro the origin. For a = b = 0, we have T d,k 0,0 = f k. But already for a = 0, b = we get a functional which is not affine invariant, but erely rotation invariant; in that case, T d,k 0, P = λ k F F F k P is the total k-diensional volue of the k-skeleton of P. In particular, T d,d 0, P is the surface area of a d-diensional polytope P R d. Finally, we ephasize that T d,d, P = dλ d P if 0 P. 8

9 If X,..., X n are n d + independent rando points in R d with X i d = N 0, I d, then P-alost surely [X i : i I] is a k-diensional polytope whenever I {,..., n} with I = k +, and we put η I := η[x i : i I], λ k,i := λ k [X i : i I], h I := h I X,..., X n. By syetry we thus obtain ET d,k a,b [X,..., X n ] = n k + h I0 η I0 a λ k,i0 b dp, where I 0 := {,..., k + }. Theore 3.4. Let X,..., X n be n d + independent rando points in R d d with X i = N 0, I d. Then, for k {0,..., d } and a, b 0, where ET d,k a,b [X,..., X n ] = n k + Cb, k, d b k k + Γ Cb, k, d := k! Γ {Y / [Y,..., Y n k ]} Y a dp, j= d+b+ j d+ j and Y, Y,..., Y n k are independent rando points in R d k with Y d = N N 0, I d k for i =,..., n k. Proof. By the sae arguents as in the proof of Theore 3., we get ET d,k 0, k+ I d d k and Y i = a,b [X,..., X n ] k+ = c 5 n, k, d... k z,..., z k+ d k+b exp z i λdz... λdz k+ R k R k i= n... gy, y k+,..., y n y a exp y i k + y R d k R d k i=k+ λ d k dy k+... λ d k dy n λ d k dy. The proof is copleted by using Lea 3. and by siplifying the constants. Clearly, a centrally syetric version of Theore 3.4 could be stated and proved in a siilar way. 4 Asyptotic expansions In Theore 3. the ean nuber of k-faces Ef k [X,..., X n ] of a Gaussian polytope in R d has been expressed in ters of a basic geoetric probability. In this section, we will derive the asyptotic expansion of probabilities of this type. For this purpose, let l, N with l + and k >, and let Y, Y,..., Y l be independent rando points in R with Y = d N 0, k + I d and Y i = N 0, I. 9

10 The choice k {0,..., d }, l = n k and = d k then corresponds to the situation of Theore 3.. In order to state our result, we define constants A, k :=, A, k :=... u,..., u R R [u,...,u ] exp u i k + u λ duλ du... λ du for, and i= C, k := +k k + Γ + k + Γ π k++ for N. An interpretation of the nubers A, k in ters of interior angles of regular siplices will be given below in the case when k N. We now consider the asyptotic behaviour of the probability that a norally distributed rando point is contained in a Gaussian polytope. Theore 4.. Let l, N and k >. Let Y, Y,..., Y l be independent rando points in R with Y = d N 0, k+ I d and Y i = N 0, I. Then as l. PY / [Y,..., Y l ] C, ka, k l k+ log l +k Cobining Theore 3. and a special case of Theore 4., we obtain the expansion.3, though with a different for of the constant, i.e. for k {0,..., d }, Ef k [X,..., X n ] Cd k, k Ad k, klog n d 4. k +! as n, giving the constant c k,d in.6. By coparison, we thus conclude that Ad k, k = d kγ d k d k π d k!k + d k d βt k, T d, 4. for k {0,..., d }. Relation 4. can be interpreted as an apparently new integral representation for the interior angles of a regular siplex. It would be nice to have a short direct proof of 4., possibly extending to ore general paraeters, if the analytic expression for βt k, T d obtained in [5,.3] with α = /d k and n = d k is used. Proof of Theore 4.. We can assue that l + and put A := {Y / [Y,..., Y l ]}. Then Wendel s theore [6] yields that l PA = PA {0 int [Y,..., Y l ]} + O. For a set F R, we define pos F := {λx : x F, λ > }; hence, if F is an -diensional convex set with 0 / F, then pos F is the truncated cone generated by F. Under the assuption that the origin is an interior point of [Y,..., Y l ], we decopose 0 l

11 the copleent of [Y,..., Y l ] into the truncated cones generated by the facets of [Y,..., Y l ]. Thus, again applying Wendel s theore and by syetry, we get l PA = P Y pos [Y,..., Y ], aff{y,..., Y } [0, Y +,..., Y l ] = l + O. Define indicator functions h 0 and h by putting and h 0 y,..., y l := {aff{y,..., y } [0, y,..., y l ] = }, h y, y,..., y := {y pos [y,..., y ]}, where y, y,..., y l R. Hence, PA can be rewritten as PA = = l k + π l+... h 0 y,..., y l h y, y,..., y R R }{{ } l+ l exp y i k + y λ dyλ dy... λ dy l + O i= l l l k + π +... φdistaff{y,..., y }, 0 l h y, y,..., y R R }{{ } + exp y i k + y λ dyλ dy... λ dy + O i= where Fubini s theore has been used in the second step. Now we first consider the case. We apply the Blaschke-Petkantschin forula. and use the rotation invariance of the integrand as in the proof of Theore 3.. Identifying R with the orthogonal copleent e R of the unit vector e, we finally get PA = pl,, k + O φ l with pl,, k := l c 6, k... φz l h y, u + ze,..., u + ze R R R u,..., u exp u i z k + y i= λ dyλ du... λ du λ dz l l l, and c 6, k := k +! Γ. π

12 It reains to evaluate the asyptotic behaviour of pl,, k. The transforation forula for ultiple integrals yields that h y, u + ze,..., u + ze exp k + y λ dy and hence = R [u,...,u ] zs exp k + s u + z λ duλ ds, l pl,, k = c 6, k... u,..., u R R [u,...,u ] exp u i I 0 l + k +, + k +, k + ; u 4.3 i= λ duλ du... λ du, where the functional I a, for a 0, was introduced in Section. Define ql,, k by the right-hand side of 4.3, but with I 0 replaced by J 0. Then Lea. iplies that PA = ql,, k + O l k+ log l +k. By substituting the definition of A, k and applying Lea., we can coplete the proof in the case. The case = follows easily by a direct arguent specializing the preceding one. 4. Again the centrally syetric case This subsection is devoted to the study of the asyptotic behaviour of the probabilities P Y / [Y,..., Y n k ] c arising in Theore 3.3. More generally, we obtain the following result by a siilar reasoning as for Theore 4.. Theore 4.. Let l, N and k >. Let Y, Y,..., Y l be independent rando points in R with Y = d N 0, k+ I d and Y i = N 0, I. Then as l. P Y / [Y,..., Y l ] c k+ C, ka, k l k+ log l +k In particular, by cobining Theores 4. and 4. we deduce the following asyptotic relation for which no direct proof sees to be known. Corollary 4.3. Let l, N and k >. Let Y, Y,..., Y l be independent rando points in R with Y = d N 0, k+ I d and Y i = N 0, I. Then as l. P Y / [Y,..., Y l ] c k+ P Y / [Y,..., Y l ]

13 Proof of Theore 4.. Assue that l. For y, y,..., y l R, we define g c y, y,..., y l := {y / [y,..., y l ] c }. Abbreviating A c := {Y / [Y,..., Y l ] c }, we get PA c = k + π l+... g c y, y,..., y l R R l exp y i k + y λ dyλ dy... λ dy l. i= Since 0 int [Y,..., Y l ] c holds P-alost surely, we can decopose R \ [Y,..., Y l ] c as in the proof of Theore 4.. Recall that h y, y,..., y = {y pos [y,..., y ]} and define h y,..., y l := {[y,..., y ] F [y,..., y l ] c }, for y, y,..., y l R. By syetry and by the reflection invariance of the noral distribution, we get PA c = l l r k + π l+... h y, y,..., y h y,..., y l r r r=0 R R l exp y i k + y λ dyλ dy... λ dy l i= l = k + π +... φdistaff{y,..., y }, 0 l R R h y, y,..., y exp y i k + y i= λ dyλ dy... λ dy. Here we used that [Y,..., Y ] is a facet of [Y,..., Y l ] c if and only if the l rando points Y +,..., Y l lie between the hyperplane aff{y,..., Y } and its reflection in the origin, P-alost surely. By the sae arguents as in the proof of Theore 4., we now obtain that PA c = l + c6, k A, k k + + O l k+ log l +k. An application of the second part of Lea. then yields the result. A cobination of Theores 3.3 and 4. and relation 4. show that φz l z exp + k + z dz Ef k [X,..., X n ] c Cd k, k Ad k, klog n d Ef k [X,..., X n ], k +! where X,..., X n is a Gaussian saple. 3

14 4. The general functional We now turn to the asyptotic expansion of the integral {Y / [Y,..., Y n k ]} Y a dp, which is related to the expected value ET d,k a,b [X,..., X n ] as shown in Theore 3.4. Again we consider a ore general situation. Theore 4.4. Let l, N and k >. Let Y, Y,..., Y l be independent rando points in R with Y = d N 0, k+ I d and Y i = N 0, I. Then {Y / [Y,..., Y l ]} Y a dp C, ka, k l k+ log l +k+a as l. Proof. We ay assue that l +. Following the proof of Theore 4., we deduce that E a l,, k := {Y / [Y,..., Y l ]} Y a dp l = c 6, k... φz l h y, u + ze,..., u + ze R R R u,..., u y a exp u i z k + y i= λ dyλ du... λ du λ dz + O φ l. By the transforation forula, h y, u + ze,..., u + ze y a exp k + y λ dy = R [u,...,u ] zs +a u + z a/ exp k + s u + z λ duλ ds. Thus, by an application of Lea. we finally get E a l,, k = l c6, k A, k k + + O l k+ log l +k+a φz l z a exp k + + z dz, fro which the assertion follows by another application of Lea.. For k {0,..., d }, we can cobine Theores 3.4 and 4.4 with 4. to find the asyptotic expansion for the expected value of the general functional ET d,k d d a,b [X,..., X n ] Cb, k, d βt k, T d π d log n d+a, 4.4 k + d 4

15 where Cb, k, d was defined in Theore 3.4. The special case a = 0, b = has already been entioned in the Introduction; the constant c k,d in.5 now follows fro 4.4. Moreover, we get here Wendel s theore is used again and Eλ d [X,..., X n ] κ d log n d EV d [X,..., X n ] ω d log n d, where κ d = ω d /d is the volue of the d-diensional unit ball. These two special relations had previously been established in []. References [] F. Affentranger. The convex hull of rando points with spherically syetric distributions. Rend. Se. Mat. Univ. Politec. Torino 49 99, [] F. Affentranger and J.A. Wieacker. On the convex hull of unifor rando points in a siple d- polytope. Discrete Coput. Geo. 6 99, [3] F. Affentranger and R. Schneider. Rando projections of regular siplices. Discrete Coput. Geo. 7 99, 9 6. [4] Y. M. Baryshnikov and R. A. Vitale. Regular siplices and Gaussian saples. Discrete Coput. Geo. 994, [5] K. Böröczky and M. Henk. Rando projections of regular polytopes. Arch. Math , [6] M. L. Eaton. Multivariate Statistics, a Vector Space Approach. Wiley, New York, 983. [7] I. Hueter. Liit theores for the convex hull of rando points in higher diensions. Trans. Aer. Math. Soc , [8] Y. Lonke. On rando sections of the cube. Discrete Coput. Geo , [9] A. M. Mathai. An Introduction to Geoetrical Probability. Gordon and Breach, 999. [0] R. E. Miles. Isotropic rando siplices. Adv. in Appl. Probab. 3 97, [] H. Raynaud. Sur l enveloppe convexe des nuages de points aléatoires dans R n I. J. Appl. Probab , [] R. Schneider. Discrete Aspects of Stochastic Geoetry. In Handbook of Discrete and Coputational Geoetry, J.E. Goodan and J. O Rourke eds, nd ed., CRC Press, Boca Raton to appear. [3] R. Schneider and W. Weil. Integralgeoetrie. Teubner, Stuttgart, 99. [4] J.W. Silverstein. The sallest eigenvalue of a large-diensional Wishart atrix. Ann. Probab , [5] A. M. Vershik and P. V. Sporyshev. Asyptotic behavior of the nuber of faces of rando polyhedra and the neighborliness proble. Selecta Math. Soviet. 99,

16 [6] J. G. Wendel. A proble in geoetric probability. Math. Scand. 96, 09. [7] J.A. Wieacker. Einige Problee der polyedrischen Approxiation. Diploarbeit, Freiburg i.br., 978. Authors addresses: Daniel Hug and Götz Olaf Munsonius, Matheatisches Institut, Albert-Ludwigs-Universität, Eckerstr., D-7904 Freiburg i. Br., Gerany e-ail: daniel.hug@ath.uni-freiburg.de, olaf@unsonius.de Matthias Reitzner, Institut für Analysis und Technische Matheatik, Technische Universität Wien, Wiedner Hauptstrasse 8 0, A-040 Vienna, Austria e-ail: Matthias.Reitzner+e4@tuwien.ac.at 6