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

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas. 2024. No. 118. P. 1–22.

Алиев А. А.

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.

Keywords: convex geometry, geometry of numbers, reverse isodiametric problem, lattice width