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.
