Given a star body $S \subset \mathbb{R}^n$ with the origin as interior point, the *critical determinant of* $S$---usually denoted as $\Delta(S)$---is the infimum of the determinants of all lattices that intersect $S$ only
at the origin.

The quantity ${\rm vol}(S)/\Delta(S)$ is a linear invariant of compact, star bodies. If we restrict its domain to the space $\mathcal{K}_0^n$ of convex bodies in $\mathbb{R}^n$ that contain the origin in their interior, then it is a continuous linear invariant.

**The question:** Is there a nice little proof or a reference for the statement that the the sublevel sets
$$
\{ K \in \mathcal{K}_0^n : {\rm vol}(K)/\Delta(K) \leq C \}
$$
are compact and hence the functional must attain a global minimum?

I think this is true (because of Macbeath's compactness theorem and the fact that $\Delta(K)^{-1}$ blows up if the origin is near the boundary of $K$), but have not really checked thoroughly thinking that is must be widely known although I cannot find the explicit statement in Cassel or Lekkerkerker.

**Addendum.** Sergei has given an excellent reason why this is not to be found in Cassel or Lekkerkerker ... However, much work has gone into giving lower bounds of the functional
$K \mapsto {\rm vol}(K)/\Delta(K)$. For example, the Minkowski-Hlawka theorem says it is bounded below by $1$ (or $\zeta(n)$ to be more accurate). *Is it known whether this functional attains a (global) minimum on* $\mathcal{K}_0^n$ ?