The Minkowski-Hlawka theorem (as stated by Gruber in his lovely book Convex and Discrete Geometry) says that if $S$ is a Jordan measurable set in $\mathbb{R}^n$ with volume < 1. Then there is a lattice $L \subset \mathbb{R}^n$ with determinant $1$ which contains no point of $S$, with the possible exception of the origin.
In the special case where the set $S$ is a star-shaped body (with respect to the origin), this inequality can be written as $$ 1 < \text{vol}(S)/\Delta(S) , $$ where $\Delta(S)$ is the critical determinant of $S$ (i.e., the infimum of the determinants of all lattices that intersect $S$ only at the origin).
Question. Does the the linear-invariant functional $S \mapsto \text{vol}(S)/\Delta(S)$ attain its greatest lower bound when defined on (1) the space of star-shaped (compact) bodies and (2) the space of convex bodies that contain the origin as an interior point.
I'm really just interested in the question of existence of minima and specially interested in the case of convex bodies. In any case, even in the "easy" case of $0$-symmetric convex bodies in $\mathbb{R}^2$, where the existence of minima is obvious, the minimum value is still just a conjecture (the Reinhardt conjecture).
