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

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 $Diam(K)\mathrm{.}$ In this paper we prove that there exists a linear bijection $T:{\mathbb{R}}^3\to {\mathbb{R}}^3$ such that $Vol(TK)\ge \frac{\sqrt{2}}{12}Diam(TK{)}^3$ 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 $Vol(K)\ge \frac{1}{12}\omega (K{)}^3$, 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 $\mathrm{\Delta }\subset {\mathbb{R}}^3$ such that $Vol(\mathrm{\Delta })=\frac{1}{12}\omega (\mathrm{\Delta }{)}^3\mathrm{.}$