Monotonicity of facet numbers of ranom convex hulls Gilles Bonnet, Julian Grote, Daniel Temesvari, Christoph Thäle, Nicola Turchi an Florian Wespi arxiv:173.31v1 [math.mg] 7 Mar 17 Abstract Let X 1,..., X n be inepenent ranom points that are istribute accoring to a probability measure on R an let P n be the ranom convex hull generate by X 1,..., X n n +1. Natural classes of probability istributions are characterize for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of P n is strictly monotonically increasing in n. Keywors. Blaschke-Petkantschin formula, mean facet number, ranom convex hull MSC. 5A, 5B5, 53C65, 6D5 1 Introuction an main result Fix a space imension. For an integer n +1, let X 1,..., X n be inepenent ranom points that are chosen accoring to an absolutely continuous probability istribution on R. By P n 1 an P n we enote the ranom convex hulls generate by X 1,..., X n 1 an X 1,..., X n, respectively. In our present text we are intereste in the mean number of facets E f 1 P n 1 an E f 1 P n of P n 1 an P n. More specifically, we ask the following monotonicity question: Is it true that E f 1 P n 1 E f 1 P n? This question has been put forwar an answere positively by Devillers, Glisse, Goaoc, Moroz an Reitzner [7] for ranom points that are uniformly istribute in a convex boy K R if = an, if 3, uner the aitional assumptions that the bounary of K is twice ifferentiable with strictly positive Gaussian curvature an that n is sufficiently large, that is, n nk, where nk is a constant epening on K. Moreover, an affirmative answer was obtaine by Beermann [4] if the ranom points are chosen with respect to the stanar Gaussian istribution on R or accoring to the uniform istribution in the -imensional unit ball for all. Beermann s proof essentially relies on a Blaschke-Petkantschin formula, a well known change-of-variables formula Faculty of Mathematics, Ruhr University Bochum, Germany. E-mail: gilles.bonnet@rub.e Faculty of Mathematics, Ruhr University Bochum, Germany. E-mail: julian.grote@rub.e Faculty of Mathematics, Ruhr University Bochum, Germany. E-mail: aniel.temesvari@rub.e Faculty of Mathematics, Ruhr University Bochum, Germany. E-mail: christoph.thaele@rub.e Faculty of Mathematics, Ruhr University Bochum, Germany. E-mail: nicola.turchi@rub.e Institute of Mathematical Statistics an Actuarial Science, University of Bern, Switzerlan. E-mail: florian.wespi@stat.unibe.ch 1
in integral geometry. Our aim in this text is to generalize her approach to other an more general probability istributions on R. In fact, we will be able to characterize all absolutely continuous rotationally symmetric istributions on R whose ensities satisfy a natural scaling property see 9 below, to which the methoology base on the Blaschke-Petkantschin formula can be applie an for which we can answer positively the monotonicity question pose above for any of these istributions. Moreover, we will apply our results to stuy similar monotonicity questions for a class of spherical convex hulls generate by ranom points on a half-sphere, which comprises as a special case the moel recently stuie by Bárány, Hug, Reitzner an Schneier [3]. To present our main result formally, we introuce four classes of probability measures: - G is the class of centre Gaussian istributions on R with ensity proportional to x exp x σ, σ >, - H is the class of heavy-taile istributions on R with ensity proportional to x 1+ x σ β, β > /, σ >, - B is the class of beta-type istributions on the -imensional centre ball B σ of raius σ with ensity proportional to x 1 x σ β, β > 1, σ >, - U comprises the uniform istributions on the 1-imensional centre spheres S 1 σ with raius σ >. It will turn out that the classes G, H, B an U contain precisely the absolutely continuous rotationally symmetric probability istributions on R, whose ensities satisfy the natural scaling property 9 below, for which monotonicity of the mean facet number of the associate ranom convex hulls can be shown by means of arguments base on a Blaschke-Petkantschin formula, see the iscussion at the en of Section 4 for further etails. In fact, our result shows that even the stronger strict monotonicity hols. Theorem 1 Let X 1,..., X n R, n + 1, be inepenent an ientically istribute accoring to a probability measure belonging to one of the classes G, H, B or U. Then, E f 1 P n > E f 1 P n 1. It shoul be emphasize that strict monotonicity of n f 1 P n cannot hol pathwise except for the trivial case n = + 1, since the aition of a further ranom point can reuce the facet number arbitrarily as the aitional point might see much more than vertices of the alreay constructe ranom convex hull. For this reason, the expectation in Theorem 1 is essential to euce a non-trivial result. We woul also like to remark that monotonicity questions relate to the volume of ranom convex hulls have recently attracte some interest in convex geometry because of their connection to the famous slicing problem. Namely, if K an L are two compact
convex sets in R with interior points, let V K an V L be the volume of the convex hull of + 1 inepenent ranom points uniformly istribute in K an L, respectively. One is intereste in the question whether the set inclusion K L implies the inequality EV K EV L. In particular, the work of Raemacher [11] shows that this is false in general whenever 4. Higher moments were treate by Reichenwallner an Reitzner [1], an we refer to the iscussion therein for further etails an backgroun material. Our text is structure as follows. In Section we recall the necessary backgroun material from integral geometry an introuce some more notation. Moreover, in Section 3 we evelop several auxiliary results that prepare for the proof of Theorem 1, which is presente in Section 4. We also iscuss in Section 4 the limitations an possible extensions of the metho we use. The final Section 5 contains the application of Theorem 1 to ranom convex hulls on a half-sphere. Backgroun material from integral geometry.1 General notation We enote by A, q the Grassmannian of all q-imensional affine subspaces of R, where q {, 1,..., }. It is a locally compact, homogeneous space with respect to the group of Eucliean motions in R. The corresponing locally finite, motion invariant measure is enote by µ q, which is normalize in such a way that µ q {E A, q : E B = } = κ q, see [14]. Here, B is the centre -imensional unit ball, κ q = π q/ is the q- Γ1+ q imensional volume of B q an Γ inicates the Gamma function. Moreover, the Beta function is given by Ba, b := 1 s a 1 1 s b 1 s = ΓaΓb, a, b >. Γa+ b We shall enote by S 1 the 1-imensional unit sphere an abbreviate by ω = κ its total spherical Lebesgue measure. For a subspace E A, q, we let λ E be the Lebesgue measure on E. For a set K R, we shall write H q K for the q-imensional Hausorff measure on K. The operators conv an span enote the convex an linear hull of the argument. We will use the notation 1 x 1,..., x to inicate the 1-imensional volume of the convex hull of points x 1,..., x.. Blaschke-Petkantschin formulae Our proof of Theorem 1 heavily relies on Blaschke-Petkantschin formulae from integral geometry. First, we rephrase a special case of the affine Blaschke-Petkantschin formula in R, which appears as Theorem 7..7 in [14]. 3
Proposition Let f : R R be a non-negative measurable function. Then, R fx 1,..., x x 1,..., x = ω 1! A, 1 H fx 1,..., x 1 x 1,..., x λ H x 1,..., x µ 1 H. Besies the affine Blaschke-Petkantschin formula in R we nee its spherical counterpart, which is a special case of Theorem 1 in [15] an can also be foun in [1, Theorem 4]. Proposition 3 Let f : S 1 R be a non-negative measurable function. Then, 1 fx 1,..., x H x S 1 1,..., x = 1! S 1 A, 1 H S 1 fx 1,..., x 1 x 1,..., x 1 h H H S 1 x 1,..., x µ 1 H, where h enotes the istance from H to the origin..3 A slice integration formula Finally, we will make use of the following special case of the spherical slice integration formula taken from Theorem A.4 in []. Proposition 4 Let f : S 1 R be a non-negative measurable function. Then, S 1 1 fxh 1 x = 1 t 3 S 1 1 S ft, 1 t yh y t. S 3 Auxiliary results 3.1 An estimate for integrals of concave functions A version of the next lemma was alreay state in [4], but without proof. For the sake of completeness we inclue here the short argument. Lemma 5 Let h : R R be a non-negative measurable function which is strictly positive on a set of positive Lebesgue measure. Further, let g : R R be a linear function with negative slope an root s [, 1]. Moreover, let L : R R be positive an strictly concave on [, 1]. Then, wherels = Ls s s. 1 hsgsls 1 s > 1 hsgsls 1 s, 1 4
Proof. We start by exploiting the positivity an strict concavity of L. For s [, s, it implies that s Ls = L s s > s s Ls, while for s s, 1], it gives Ls < s s Ls. 3 Since g has a negative slope, it is positive on[, s an negative ons, 1]. Splitting the integral on the left han sie of 1 at the point s an using an 3, respectively, yiels 1 = > = hsgsls 1 s s s 1 hsgsls 1 s+ 1 s hsgsls 1 s s 1 1 s 1 hsgs s Ls s+ hsgs s Ls s s hsgsls 1 s. This completes the argument. 3. Computation of marginal ensities Recall the efinitions of the istribution classes H, B an U. It will turn out that it suffices to consier the cases where the scale parameters σ there are equal to 1. From now on we restrict to these cases an enote the ensity of a istribution in H by p H,β x = π / Γβ Γβ β, x R, β > /, 4 1+ x that of a istribution in B by p B,β x = π / Γ + β+1 1 x β, x B, β > 1, Γβ+ 1 an note that the uniform istribution on S 1 has ensity p U x = 1 ω, x S 1, with respect to the spherical Lebesgue measure. The next lemma provies formulas for the ensities of the one-imensional marginals of these istributions an shows, that the classes B an H are in some sense close uner one-imensional projections. Since all istributions we consier are rotationally symmetric, it is sufficient to consier projections onto the first coorinate. We woul like to emphasize that the proof of Lemma 6 uses in an essential way the scaling property 9 below of the involve ensities. 5
Lemma 6 Let Π : R R,x 1,..., x x 1 be the projection onto the first coorinate. i Let P H be a istribution with ensity p H,β for some β > /. Then, the image measure of P uner Π has ensity Γ β 1 f H,β x = π 1/ 1+ x 1 β, x R. Γ β ii Let P B be a istribution with ensity p B,β for some β > 1. Then, the image measure of P uner Π has ensity f B,β x = Γ β+1+ π 1/ 1 x 1 +β, x [ 1, 1]. Γ f U x = π 1/ β+ +1 iii Let P U be the uniform istribution on S 1. Then, the image measure of P uner Π has ensity Γ Γ 1 1 x 3, x [ 1, 1]. Proof. To prove i we put x = x 1,..., x R, y := x,..., x an also efine c H,,β := π / Γβ Γβ /. Then, R 1 c H,,β = 1+ x β x,..., x R 1 c H,,β 1+ x 1 β = 1+ x 1 β R 1 c H,,β = 1+ x 1 1 β c H,,β H,,β c H, 1,β 1+ x 1 1 β, β y 1+ y 1+ x1 1+ z β 1+ x 1 1 z c c H, 1,β H, 1,β R 1 1+ z β z where we use the substitution z = y/ 1+ x1. Plugging in the constants yiels the esire result. Next, we consier the istribution with ensity p B,β. For x = x 1,..., x B, we put again y := x,..., x an abbreviate c B,,β := π / Γ +β+1. Then, similarly as Γβ+1 6
above, we compute B 1 c B,,β = 1 x β x,..., x B 1 c B,,β 1 x 1 β = 1 x 1 β B 1 c B,,β = 1 x 1 1 +β c B,,β B,,β c B, 1,β 1 x 1 1 +β, β y 1 y 1 x1 1 z β 1 x 1 1 z c c B, 1,β B, 1,β B 1 1 z β z where we use the substitution z = y/ 1 x1. Again, simplification of the constants yiels the esire result. Finally, we consier the case of the uniform istribution on S 1. We enote by F the istribution function of the image measure of P = ω 1 H 1 uner the orthogonal S 1 projection map Π an let x 1 [ 1, 1]. Using the slice integration formula from Proposition 4, we obtain Fx 1 = 1 { } H 1 u S 1 : Πu [ 1, x ω S 1 1 ] = 1 1{Πu [ 1, x 1 ]}H 1 u ω S 1 S 1 = 1 ω x 1 1 = ω 1 ω 1 t 3 x 1 1 S 1 t 3 t. H S y t Differentiation with respect to x 1, together with the efinitions of ω an ω 1, complete the proof. In what follows, we shall enote by F H,β, F B,β an F U the istribution functions corresponing to the ensities f H,β, f B,β an f U compute in Lemma 6, respectively. Remark 7 The marginal ensities of the Gaussian istributions belonging to the class G can also be compute along the lines of the proof of Lemma 6. This yiels oneimensional Gaussian marginals. Since ranom convex hulls of Gaussian points have alreay been treate in [4], we ecie to concentrate on the classes H, B an U. 7
4 Proof of Theorem 1 an iscussion 4.1 Proof of Theorem 1 Base on the results from the two previous sections we are now able to present the proof of our main result. In what follows, we enote all constants by c. Unless otherwise state, they only epen on the space imension an the parameter β of the unerlying probability istribution. Their value might change from instance to instance. Proof of Theorem 1. For the classes G, H, B an U it is sufficient to consier the case that the scale parameter σ is equal to 1, since the mean facet number is invariant uner rescalings. We start with the istributions from class G. However, since this case has alreay been treate in [4], we refer to Theorem 5.3.1 there. Next, we consier the heavy-taile istribution on R with ensity p H,β x H,,β 1+ x β, where β > / an c H,,β = π / start with the equality Γβ Γβ /. We follow the ieas from [4] an E f 1 P n = E = 1 i 1 <...<i n 1{convX i1,..., X i is a facet of P n } n PconvX 1,..., X is a facet of P n, 5 which hols ue to the fact that the ranom points X 1,..., X n are inepenent an ientically istribute. Let us enote by H A, 1 the affine hull of the 1- imensional simplex P spanne by X 1,..., X. In the case that P is a facet of P n, all the remaining points X +1,..., X n have to lie in one of the open half-spaces etermine by H. If we enote by Π H the orthogonal projection onto H, the orthogonal complement of H, we observe that P is a facet of P n if an only if the point Π H P is not containe in the interior of the interval Π H P n on H. Therefore, using Lemma 6, the affine Blaschke-Petkantschin formula from Proposition an the abbreviation F = F H,β Π H P, we get for the probability that P is a facet of P n, PconvX 1,..., X is a facet of P n = 1 F n +F n c H,,β 1+ x i β x 1,..., x i=1 R A, 1 H 1 F n +F n 1 x 1,..., x λ H x 1,..., x µ 1 H. i=1 c H,,β 1+ x i β Next, we use the theorem of Pythagoras to ecompose, for each i {1,..., }, the norm x i. Namely, writing H for the Eucliean norm in H A, 1 an h for the istance from H to the origin in R, we have that x i = x i H + h. 8
Therefore an as alreay use in the proof of Lemma 6, the last term of the integran can be rewritten as 1+ x i β = 1+h + x i H β = 1+h 1+ β x i β H 1+ h. 6 Moreover, since each hyperplane H = Hu, h is uniquely etermine by its unit normal vector u S 1 an its istance h [, to the origin, the integration over A, 1 can be replace by a twofol integral over S 1 an [,. Using the substitutions y i = x i / 1+h with λ H x i = 1+h 1/ λ H y i, the rotational invariance of the unerlying probability measure, an writing Fh for F H,β h as well as fh for f H,β h, gives in view of Lemma 6 that PconvX 1,..., X is a facet of P n S 1 S 1 H 1 Fh n + Fh n 1 x 1,..., x 1+h β c H,,β 1+ x i β H 1+h i=1 λ H x 1,..., x hh 1 u S 1 1 Fh n + Fh n 1+h 1 β + 1 hh 1 u S 1 β 1 y 1,..., y c H, 1,β 1+ y i H λ H y 1,..., y H i=1 1 Fh n + Fh n 1+h β 1 + 1 h 1 Fh n fh 1+h 1 h, where we also use the fact that the integral over H is a finite constant given by Equation 7 in [1] an which only epens on the space imension an on β. Write now s = Fh an Ls = f F 1 s 1+F 1 s to obtain 1 PconvX 1,..., X is a facet of P n 1 s n Ls 1 s. Thus, combination of the above computation with 5 yiels the representation E f 1 P n E f 1 P n 1 1 [ ] n n 1 1 s 1 s n 1 Ls 1 s. In orer to apply Lemma 5, we have to verify that Ls is strictly concave on, 1. We prove this by showing that the secon erivative of Ls is negative. So, let c H,1,β := 9 7
Γβ π 1/ Γβ 1/ an recall that fx H,1,β1+ x 1 β from Lemma 6. Furthermore, from the efinition of F it follows that F 1 1 s = f F 1 s = 1. 8 1 c H,1,β 1+F 1 β s We recall that Ls = f F 1 s 1+F 1 s H,1,β 1+F 1 s β. Hence, using 8, the first erivative of Ls is L s H,1,β β 1+F 1 s β F 1 s F 1 s = β 1+F 1 s 1 F 1 s an, thus, for the secon erivative we fin that [ L s = β 1+F 1 s 1 F 1 s 1 1+F 1 s 3 ] F 1 s F 1 s = = c H,1,β = c H,1,β <, [ β F 1 s 1+F 1 s 1 β 1+F 1 s β 1 [ 1+F 1 s F 1 s ] 1+F 1 s β 1 β 1+F 1 s 3 ] F 1 s where the last inequality follows from the fact that β > /. As a consequence, we can apply Lemma 5 to euce that E f 1 P n E f 1 P n 1 1 [ ] n n 1 1 s 1 s n 1 Ls 1 s L/n > /n L/n = /n =, 1 1 n 1 n 1 s n 1 s 1 1 s n n B, n +1 n n B, n s where we use the well-known relation B, n +1 = n n B, n for the beta function. 1
As the next case we consier the class B of beta-type istribution on the unit ball B with ensity f B,β for some β > 1. In this case the proof follows almost line by line the proof for H, up to some minor moifications. In particular, 7 stays the same except that now Ls = f F 1 s 1 F 1 s, where Fh = F B,β h an fh = f B,β h. Therefore, it follows that L s = β+ 1 F 1 s β 1, c B,1,β where the constant c B,1,β is c B,1,β := π 1/ Γβ+ 3 Γβ+ 1 1. Since F 1 s 1, 1, we obtain L s < an can conclue as in the proof for the class H presente above. Finally, we consier the case of the uniform istribution on S 1. Here we get by applying the spherical Blaschke-Petkantschin formula from Proposition 3 an using the abbreviations Fh = F U h an fh = f U h, PconvX 1,..., X is a facet of P n 1 Fh n + Fh n 1 x 1,..., x 1 h A, 1 H S 1 S 1 S 1 1 1 H H S 1 x 1,..., x µ 1 H H S 1 1 Fh n + Fh n 1 x 1,..., x 1 h H H S 1 x 1,..., x hh 1 S 1 u 1 Fh n + Fh n 1 h + 1 hh 1 S 1 u 1 y 1,..., y H y S 1,..., y, S where the substitution x i = y i 1 h with H H S 1 x i = 1 h H H S 1 y i was use. In particular, this transforms the integration over H S 1 into a -fol integral over the unit sphere in H, which in turn has been ientifie with S ue to rotational invariance. Since the integral over S is a known positive constant only epening on the precise value can be euce from [14, Theorem 8..3], for example, we get by rotational invariance of the unerlying istribution that PconvX 1,..., X is a facet of P n 1 1 1 1 Fh n + Fh n 1 h 3 + 1 h 1 Fh n fh 1 h 1 h. 11
As a consequence, also for the uniform istribution on S 1 we arrive at an expression of the form 7, this time with Ls = ff 1 s 1 F 1 s. From this point on, the proof can be complete as in the case of the istribution class H or B. This completes the argument. 4. Discussion Let p : R [, enote a probability ensity. By a careful inspection of the proof of Theorem 1 we see that the following properties of the ensity p have been use there. First of all, we use that p is spherically symmetric, that is, px only epens on x = x 1,..., x R via x. By abuse of notation, we shall write pr :, [, with r = x1 +...+x for the raial part of the ensity p. This was essential to apply the Blaschke-Petkantschin formulae, which use the invariant hyperplane measure µ 1. Moreover, given H A, 1 with istance h to the origin, we have use that we can fin ϕh, ψh > such that p r r + h = ϕh p 9 ψh for all r >. For example, for the ensity p H,β, β > /, the scaling property 9 is satisfie with ϕh = 1+h β an ψh = 1+ h, see 6. This scaling property was essential when we separate what happens within H from the contribution that arises from the istance of H to the origin. However, all rotationally symmetric ensities with almost everywhere ifferentiable raial part satisfying the scaling property 9 with an almost everywhere ifferentiable function ψ have been classifie by Miles [1] see p. 376 there an Ruben an Miles [13]. They precisely correspon to the istributions in the classes G, H, B as well as to the exceptional istributions in U, for which Theorem 1 is formulate. On the other han, this oes not mean that G, H, B an U contain the only rotationally symmetric istributions on R for which such computations are possible. For example, the ensity with raial part p β,j r β,,j r j /1+r β, r >, j {, 1,,...} an β > j+/, which oes not belong to the class H whenever j >, satisfies the following generalization of the scaling property 9: with ϕ k h = p β,j r + h = j r ϕ k hp β,k ψh k= j h k j 1+h β k =,..., j an ψh = 1+ h k. One can check that the 1-imensional marginal ensity of p β,j equals f β,j x 1 = j k= j cβ,,j x k j k c 1 1+ x 1 1 k+ β, β, 1,k an that from here on the argument base on the affine Blaschke-Petkantschin formula can be applie term-by-term. Unfortunately, the computations in such an similar situations become quite involve. Moreover, to classify all rotationally symmetric ensities on R for which these computations can be performe seems to be out of reach. 1
One might also ask whether the metho base on Blaschke-Petkantschin formulae yiels monotonicity of the mean facet number in such situations where the ranom points X 1,..., X n are inepenent with istributions belonging to one of the classes G, H, B an U, but not necessarily the same a so-calle mixe case. That is, some of the X i s are Gaussian, some istribute accoring to a istribution in H etc. but within each class we choose every time the same scale parameter σ. Unfortunately, this oes not work an, in fact, the metho breaks own. The reason is that each istribution class requires its iniviual substitution, which is aapte to its respective scaling property 9. The resulting ifferent rescalings in the hyperplane H istort the relationship between the 1-volume in H before an after the transformation, cf. [13]. 5 Ranom convex hulls on a half-sphere In this section we consier an application of Theorem 1 to convex hulls generate by ranom points on a half-sphere. We fix, enote by S the -imensional unit sphere in R +1 an efine the half-sphere S + = {y = y 1,..., y +1 S : y +1 > }. Furthermore, we let S be the class of probability istributions on S + that have ensity p S,α y S,α y α +1, y = y 1,..., y +1 S +, α > 1, with respect to the spherical Lebesgue measure on S +. Here, c S,α > is a suitable normalization constant. In particular, choosing α = shows that the uniform istribution on S + belongs to the class S. For fixe α > 1 an n + 1 we let X 1,..., X n be inepenent ranom points that are istribute on S + accoring to the ensity p S,α. By S n we enote the spherical convex hull of X 1,..., X n, that is, the smallest spherically convex set in S + containing the points X 1,..., X n. For the special choice α =, this moel has recently been stuie in [3]. In particular, it is shown in [3] that for this choice of α the mean number of facets E f 1 S n of the spherical ranom polytope S n converges to a finite constant only epening on, as n a similar result is in fact vali for all istributions in S, see [1, 6, 8]. As a special case, our next result shows the somewhat surprising fact that this limit is approache in a strictly monotone way. Theorem 8 Let X 1,..., X n, n +1, be inepenent an ientically istribute accoring to a probability measure belonging to the class S. Then, E f 1 S n > E f 1 S n 1. Proof. Let g : R S + be the mapping efine as x gx = 1,..., 1+ x x 1+ x, 1 1+ x, with inverse given by g 1 y = y1 y +1,..., y y +1 13
this is known as the gnomonic projection. Let Dg be the Jacobian matrix of g an put J g x := et Dgx Dgx. Then, it hols that J g x = 1+ x +1, see [5, Proposition 4.]. Moreover, for a measurable subset A R an a measurable function f : A R the area formula [9, Theorem 3..3] says that A fx x = ga f g 1 yj g g 1 y 1 H S +y. Next, we notice that 1+ g 1 y = y +1 an apply the formula with fx = p H,βx for some β > /: A c H,,β 1+ x β x = ga c H,,β y β 1 +1 H S +y, where c H,,β = π / Γβ/Γβ is the normalization constant of the ensity p H,β efine in 4. As a result, we see that the ensity p S,β 1 on S + is the push-forwar of the ensity p H,β on R uner g. Note also that β 1 > 1 since β > / an that the uniform measure on the half-sphere correspons to the choice β = + 1/. The above iscussion shows the following. Let P n be the ranom convex hull in R generate by n inepenent points with ensity p H,β. Then, the push-forwar of P n has the same istribution as the spherical ranom polytope S n with α = β 1. Moreover, the facets of P n are in one-to-one corresponence with those of S n. As a consequence, the mean facet number of the spherical ranom polytope S n is the same as the mean facet number of the ranom convex hull P n, i.e., E f 1 S n = E f 1 P n. Thus, the monotonicity follows from Theorem 1. Acknowlegement We woul like to thank Zakhar Kabluchko Münster for a number of useful conversations an for bringing references [8] an [13] to our attention. Thanks go also to Matthias Reitzner Osnabrück for his valuable remarks on an earlier version of this text. JG, DT an NT have been supporte by the Deutsche Forschungsgemeinschaft DFG via RTG 131 High-imensional Phenomena in Probability Fluctuations an Discontinuity. References [1] Alous, D., Fristet, D., Griffin, P. an Pruitt, W.: The number of extreme points in the convex hull of a ranom sample. J. Appl. Probab. 8, 87 34 1991. [] Axler, S., Bouron, P. an Ramey, W.: Harmonic Function Theory. Volume 137 of Grauate Texts in Mathematics, Springer, New York, secon eition 1 14
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