# A question on the Mahler conjecture

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and $$K^* := \{y \in \mathbb{R}^n : \langle y, x \rangle \leq 1 \mbox{ for all } x \in K \}$$ is its polar body, then the product of volumes $|K| |K^*|$ is bounded below by $(n+1)^{n+1}/(n!)^2$. Equality is conjectured to hold only for simplexes.

Is it known whether there is a unique minimum of $K \mapsto |K| |K^*|$ modulo linear equivalence?

Is there some dimension dependent bound on the number of minima (modulo linear equivalence)?

For a given dimension is it least known that the number of minima (modulo linear equivalence) is finite?

This last would follow if it were known that the minima or local minima are isolated points in the space of linear equivalence classes of convex bodies containing the origin as an interior point. I guess that's only known for the explicit case of the simplex.

I know that determining whether the minima are polytopes is still open (and that in the symmetric case there are different linearly inequivalent classes of conjectured minima), but I don't remember having seen the problem of uniqueness discussed in the asymmetric case.

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I'm pretty sure that no such partial results are known, not because I'm an expert on the subject but because I've heard a lot of talks (including quite recently) on the subject and I've never heard any such results. Also in the symmetric case, it's believed that Hanner polytopes are the only minimizers, but I've never heard of any results bounding the number of minima. –  Mark Meckes Apr 29 '13 at 17:09
Yes, I also haven't heard anything and sometimes it seems to me that everyone around me is trying to prove this conjecture (without always acknowledging it), but it can't hurt to ask. I'm always surprised by the input one gets from MO. –  alvarezpaiva Apr 29 '13 at 17:13