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

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 $Vol(\Delta) = \frac{1}{12}\omega(\Delta)^3.$