Compact convex sets of the plane and probability theory

TL;DR

This paper explores the relationship between probability measures and convex sets in the plane, providing explicit formulas, operations translation, convergence results, and models for random convex sets.

Contribution

It introduces explicit formulas linking convex set borders to probability measures and translates operations like Minkowski sum into measure operations.

Findings

Minkowski sum corresponds to a natural measure operation.

Sampled polygonal curves converge to a convex set at rate √n.

Models and simulations of smooth random convex sets are developed.

Abstract

The Gauss-Minkowski correspondence in $\mathbb{R}^2$ states the existence of a homeomorphism between the probability measures $\mu$ on $[0,2\pi]$ such that $\int_0^{2\pi} e^{ix}d\mu(x)=0$ and the compact convex sets (CCS) of the plane with perimeter~1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS -- for example, the Minkowski sum -- have natural translations in terms of probability measure operations, and reciprocally, the convolution of measures translates into a new notion of convolution of CCS. Additionally, we give a proof that a polygonal curve associated with a sample of $n$ random variables (satisfying $\int_0^{2\pi} e^{ix}d\mu(x)=0$) converges to a CCS associated with $\mu$ at speed $\sqrt{n}$, a result much similar to the convergence of the empirical process in statistics. Finally, we employ this correspondence to present models of smooth random CCS and simulations.