A Uniform Asymptotical Upper Bound for the Variance of a Random Polytope in a Simple Polytope

Let X be a convex body in R. A random polytope Pn(X) is a convex hull of n independent uniformly distributed points in X. Starting with the paper by Rényi and Sulanke [3], random polytopes have been a popular object for research in stochastic geometry. There are numerous results concerning different functionals of Pn(X), and the most important of these functionals are the volume vol Pn(X) and the components of the f -vector fi(Pn(X)). Since 1990s a lot of research has been done on the distributional properties of stochastic variables of type A(Pn(X)), where A is a given functional of a polytope. Most of this research uses different estimates for Var A(Pn(X)). In [2] Wieacker and Weil state that the determination of the variance is a major open problem. Let X be a polytope. The paper [4] estimates Var fd−1(Pn(X)) as follows: E fd−1(Pn(X)) ¿ Var fd−1(Pn(X)) ¿ C(X) E fd−1(Pn(X)),

The present paper contains a sketch of the proof of an upper bound for the variance of the number of hyperfaces of a random polytope when the mother body is a simple polytope.Thus we verify a weaker version of the result in [1] stated without a proof.The article is published in the author's wording.
Let X be a convex body in R d .A random polytope P n (X) is a convex hull of n independent uniformly distributed points in X.
Starting with the paper by Rényi and Sulanke [3], random polytopes have been a popular object for research in stochastic geometry.There are numerous results concerning different functionals of P n (X), and the most important of these functionals are the volume vol P n (X) and the components of the f -vector f i (P n (X)).
Since 1990s a lot of research has been done on the distributional properties of stochastic variables of type A(P n (X)), where A is a given functional of a polytope.Most of this research uses different estimates for Var A(P n (X)).In [2] Wieacker and Weil state that the determination of the variance is a major open problem.
Let X be a polytope.The paper [4] estimates Var f d−1 (P n (X)) as follows: where C(X) depends on the combinatorial structure of X.However, it seems to be natural to change the bounds so that their ratio would be independent of X.
We obtain an inequality with coincident upper and lower bounds in the case of a simple polytope X. Namely, we prove the following theorem Theorem 1.There exist positive real numbers C 1 , C 2 such that for every simple polytope X and every positive integer n > n 0 (X) one has Consider a Poisson polytope Π λ (X), i.e. the convex hull of a Poisson process in X of intensity λ.If we prove the inequality for λ > λ 0 (X), then theorem 1 will immediately follow from the result [5] by J. Pardon.So, our goal is to prove (1).For a generic d-tuple x = (x 1 , x 2 , . . ., x d ) ∈ X d the hyperplane aff(x 1 , x 2 , . . ., x d ) is uniquely defined and splits X into 2 caps.Denote by C(x) the smaller of those caps.Define Let S ⊂ X d be a measurable set invariant under all permutations of the d-tuple.Say that a facet of Π λ (X) corresponds to S if the d-tuple of its vertices belongs to S and the smaller cap defined by its affine hull contains no vertex of Π λ (X).Denote by f d−1 (S) the number of facets of Π λ (X) corresponding to S. Let a i (S) be the number of ordered pairs (F 1 , F 2 ) such that F 1 , F 2 are facets of Π λ (X) corresponding to S with exactly i common vertices.

No point z j belongs to C(x(z)) ∪ C(y(z)).
According to Slivnyak-Mecke formula (see, for example [6]), By the same formula, Applying to (4) the identity Моделирование и анализ информационных систем Т. 19, № 6 (2012) Now we can rewrite the last term of (2) as follows: Finally, we have obtained the integral expressions for all the terms of (2).Suppose {S 1 , S 2 , . . ., S N } is a partition of X d .Then we can observe that where fd−1 is the number of facets of Π λ (X) such that a greater cap of this facet contains no vertex of Π λ (X).Our goal now is to construct a nice partition of X d .Recall that X is a simple polytope.Enumerate all the faces of X: F 1 , F 2 , . . .F N so that i > j whenever dim F i > dim F j .(We treat X as its own face, so F N = X.) For every generic d-tuple x ∈ X d find the face F i with the 2 properties: 1. dim aff(vert 2. i is the maximal number to satisfy condition 1.
Let S i be the set of all d-tuples corresponding to F i .Write down all the estimates we need to proceed.
and F j are vertices with no adjoining edge.

For all other covariations apply the inequality
Writing down Var f d−1 (Π λ (X)) as the variance of the sum (7), we see that where M is the number of vertices of X.This gives the upper bound for the variance.The lower bound is due to Bárány and Reitzner (see [4]).
Remark.Probably, the analogue of Theorem 1 is true for all convex polytopes X (and this is exactly the statement left in [1] without a proof).However, the author was not able to get a satisfactory uniform estimate for Var f d−1 (S i ) when F i is a vertex.We can go even further and conjecture the analogue of Theorem 1 for all convex bodies.However, the methods described above are not applicable in the most general case.