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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 $\mathcal{K}_0^n$ ?

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up vote 6 down vote accepted

It is not compact unless you allow the origin at the boundary (or maybe impose some kind of uniform strict convexity).

Consider a rectangle $K=[-1,1]\times[-\delta,1]$ in the plane, where $\delta$ is positive and goes to 0. If $K$ intersects some lattice only at the origin then so does $-K$, by symmetry. But $K\cup -K$ is a convex symmetric set of area 4, hence $\Delta(K)=\Delta(K\cup -K) \ge 1$ by Minkowski's theorem.

Therefore the ratio $vol/\Delta$ is bounded for all such $K$, but they have no limit in $\mathcal K^2_0$ as $\delta\to 0$.

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Hi Sergei, you're right. I had some kind of uniform convexity in mind when saying that $\Delta(K)^{-1}$ blows up as the origin moves close to the boundary. Thanks ! –  alvarezpaiva Mar 25 '13 at 15:30
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