In its asymmetric version, the [Mahler conjecture](https://en.wikipedia.org/wiki/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. 



A question on the Mahler conjecture
No, it is not known that the minimum is unique, but it is believed to be. In fact, [this paper by Kim and Reisner](http://arxiv.org/abs/1001.0217) proves that the simplex is (modulo linear equivalence) a strict local minimum; thus the whole conjecture would follow from uniqueness of *local* minima.


