A Lower Bound on the Waist of Unit Spheres of Uniformly Convex Normed Spaces

TL;DR

This paper establishes a new lower bound on the waist of unit spheres in uniformly convex normed spaces using advanced topological and geometric techniques, extending previous isoperimetric inequalities.

Contribution

It introduces a novel lower bound on the waist of spheres in uniformly convex spaces, generalizing existing isoperimetric inequalities with new localization and topological methods.

Findings

Provides a new lower bound on the waist of unit spheres in uniformly convex spaces

Extends Gromov-Milman isoperimetric inequality to broader settings

Utilizes advanced localization and Borsuk-Ulam techniques

Abstract

In this paper we give a lower bound on the waist of the unit sphere of a uniformly convex normed space by using the localization technique in codimension greater than one and a strong version of the Borsuk-Ulam theorem. The tools used in this paper follow ideas of M. Gromov in [4]. Our isoperimetric type inequality generalizes the Gromov-Milman isoperimetric inequality in [5].