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).