No return to convexity

TL;DR

This paper investigates how classes of log-concave measures behave under limits and transformations, showing that starting from one-dimensional measures does not produce non-trivial uniform measures on convex bodies.

Contribution

It proves that the closure of one-dimensional log-concave measures under certain operations does not include non-trivial uniform measures on convex bodies.

Findings

No non-trivial uniform measures on convex bodies are obtained from one-dimensional log-concave measures.

The study characterizes the limits of log-concave measures under transformations.

The results clarify the structure of log-concave measures and their limitations.

Abstract

In the paper we study closures of classes of log--concave measures under taking weak limits, linear transformations and tensor products. We consider what uniform measures on convex bodies can one obtain starting from some class $\mathcal{K}$. In particular we prove that if one starts from one--dimensional log--concave measures, one obtains no non--trivial uniform mesures on convex bodies.