Random zonotopes and valuations

TL;DR

This paper introduces a model of random zonotopes formed by summing independent random segments, computes their expected intrinsic volumes and valuations, and establishes a central limit theorem for these valuations.

Contribution

It defines a new model of random zonotopes, derives formulas for expected valuations, and proves a central limit theorem for these valuations.

Findings

Expected intrinsic volumes are derived under mild distribution assumptions.

A central limit theorem is established for valuations of the random zonotopes.

The work extends valuation theory to a probabilistic setting for zonotopes.

Abstract

We define a random zonotope in Euclidean space, by adding finitely many random segments, which are independently and identically distributed. For this random polytope, we determine, under a mild assumption on the distribution, the expectations of the intrinsic volumes, more generally, the expectations of suitable valuations. We also prove a central limit theorem for a valuation evaluated at these random zonotopes.