On the reverse isodiametric problem and Dvoretzky-Rogers-type volume bounds

TL;DR

This paper investigates a reverse isodiametric problem, establishing bounds and solutions for symmetric convex bodies and connecting it to volume bounds for simplices via matrix decomposition techniques.

Contribution

It solves the reverse isodiametric problem for symmetric convex bodies and derives new Dvoretzky-Rogers-type volume bounds using matrix minors.

Findings

Solved the reverse isodiametric problem for o-symmetric convex bodies.

Established asymptotic bounds for the general reverse isodiametric problem.

Derived new volume bounds for simplices using matrix analysis techniques.

Abstract

The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the reverse question: Every convex body has a linear image whose isodiametric quotient is at least as large as that of a regular simplex. We relate this reverse isodiametric problem to minimal volume enclosing ellipsoids and to the Dvoretzky-Rogers-type problem of finding large volume simplices in any decomposition of the identity matrix. As a result, we solve the reverse isodiametric problem for $o$-symmetric convex bodies and obtain a strong asymptotic bound in the general case. Using the Cauchy-Binet formula for minors of a product of matrices, we obtain Dvoretzky-Rogers-type volume bounds which are of independent interest.