The Alon--Milman Theorem for non-symmetric bodies

TL;DR

This paper extends the classical Alon--Milman theorem, which concerns symmetric convex bodies, to the broader class of non-symmetric convex bodies, revealing similar projection properties.

Contribution

The authors generalize the Alon--Milman theorem from symmetric to non-symmetric convex bodies, broadening its applicability in convex geometry.

Findings

Extended the theorem to non-symmetric bodies

Identified large-dimensional projections close to Euclidean balls or cross-polytopes

Provided new bounds for projections of convex bodies

Abstract

A classical theorem of Alon and Milman states that any $d$ dimensional centrally symmetric convex body has a projection of dimension $m\geq e^{c\sqrt{\ln{d}}}$ which is either close to the $m$-dimensional Euclidean ball or to the $m$-dimensional cross-polytope. We extended this result to non-symmetric convex bodies.