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

A Magazinov

doi:10.18255/1818-1015-2012-6-148-151

Abstract

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.

Highlights

Starting with the paper by Renyi and Sulanke [3], random polytopes have been a popular object for research in stochastic geometry
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
We obtain an inequality with coincident upper and lower bounds in the case of a simple polytope X

Summary

Introduction

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)). We obtain an inequality with coincident upper and lower bounds in the case of a simple polytope X. There exist positive real numbers C1, C2 such that for every simple polytope X and every positive integer n > n0(X) one has

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Conclusion