Convex bodies generated by sublinear expectations of random vectors

TL;DR

This paper unifies various convex bodies in geometry by showing they can be generated through sublinear expectations of random vectors, revealing a common underlying construction and duality.

Contribution

It introduces a general framework linking convex bodies with sublinear expectations, providing a unified perspective and dual representation for these geometric objects.

Findings

Many convex bodies are special cases of a general sublinear expectation construction.

A dual representation of these convex bodies is identified.

A foundational construction for these bodies is described.

Abstract

We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of a general construction, relying on applying sublinear expectations to random vectors in Euclidean space. We identify the dual representation of such convex bodies and describe a construction that serves as a building block for all so defined convex bodies.