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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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  • $\begingroup$ 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. $\endgroup$
    – Mark Meckes
    Commented Apr 29, 2013 at 17:09
  • $\begingroup$ 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. $\endgroup$
    – alvarezpaiva
    Commented Apr 29, 2013 at 17:13

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No, it is not known that the minimum is unique, but it is believed to be. In fact, this paper by Kim and Reisner proves that the simplex is (modulo linear equivalence) a strict local minimum; thus the whole conjecture would follow from uniqueness of local minima.

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  • $\begingroup$ Thanks Mark. Yes, I know that paper. I am just curious as to whether there are non-trivial partial results in this direction. $\endgroup$
    – alvarezpaiva
    Commented Apr 29, 2013 at 15:42
  • $\begingroup$ Something like there is only a finite number of minima up to linear equivalence. Note to MO: it would be nice to be able to edit comments ... $\endgroup$
    – alvarezpaiva
    Commented Apr 29, 2013 at 15:45
  • $\begingroup$ @Mark: you're right, it seems no one knows anything about this! $\endgroup$
    – alvarezpaiva
    Commented Jun 3, 2013 at 8:23

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