Convex subsets of non-convex Lorentz balls

TL;DR

This paper investigates the properties of convex subsets within non-convex Lorentz balls, providing bounds on Gaussian measure distributions by approximating sub-level sets with convex regions, inspired by phenomena observed in star bodies.

Contribution

It introduces bounds on Gaussian measure distributions for functions related to Lorentz spaces using convex approximations of sub-level sets, connecting to the randomized Dvoretzky theorem.

Findings

Bounds on Gaussian measure distributions derived

Convex approximations effectively model non-convex Lorentz balls

Connections established with the randomized Dvoretzky theorem

Abstract

Many star bodies have convex subsets with approximately the same Gaussian measure (of the complement). Inspired by this phenomenon, and in connection with the randomized Dvoretzky theorem for Lorentz spaces, we derive bounds on the distribution of certain functions of a Gaussian random vector by approximating their sub-level sets by convex subsets.