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 $T:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}^{3}$ such that Vol$(TK)\ge \frac{\sqrt{2}}{12}\text {Diam}(TK)^3$ with equality if K is a simplex, which was conjectured by Makai Jr. (Studia Sci Math Hungar 13:19–27, 1978) (see also Behrend (Math Ann 113:713–747, 1937. https://doi.org/10.1007/BF01571662). As a corollary, we prove that any set of non-separable translates in a lattice 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. Moreover, this estimate is tight for some simplex.