The exact bound for the reverse isodiametric problem in 3-space

TL;DR

This paper proves the exact volume bound for convex bodies in 3D related to their diameter, confirming a conjecture, and explores implications for lattice packings and convex geometry.

Contribution

It establishes the precise volume bound for convex bodies in 3D under linear transformations, confirming Makai Jr.'s conjecture and deriving related lattice packing bounds.

Findings

Existence of a linear transformation achieving the volume bound.

Confirmation of the conjectured volume bound for simplices.

Lower bound on lattice packing density in 3D.

Abstract

Let $K$ be a convex body in $\mathbb{R}^{3}$. We denote the volume of $K$ by $Vol(K)$ and the diameter of $K$ by $Diam(K).$ In this paper we prove that there exists a linear bijection $T:\mathbb{R}^{3}\to \mathbb{R}^{3}$ such that $Vol(TK)\geq \frac{\sqrt{2}}{12}Diam(TK)^3$ with equality if $K$ is a simplex, which was conjectured by Endre Makai Jr. As a corollary, we prove that any non-separable lattice of translates in $\mathbb{R}^{3}$ has density of at least $\frac{1}{12}$, which is a dual analog of Minkowski's fundamental theorem. Also we prove that $Vol(K)\geq \frac{1}{12}\omega(K)^3$, where $K\subset \mathbb{R}^{3}$ is a convex body and $\omega(K)$ is the lattice width of $K$. In addition, there exists a three-dimensional simplex $\Delta\subset \mathbb{R}^3$ such that $Vol(\Delta) = \frac{1}{12}\omega(\Delta)^3.$