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It seems that I have unwittingly started an “open problem of the week” column here; certainly it seems easier for me to pose unsolved problems than to write papers :-) .

This question in convex geometry has been around for a while; I am fond of it because it attempts to capture the intuitively obvious fact that cubes and octahedra are the “pointiest” possible symmetric convex bodies one can create. Sadly, we still have very few tools to make this intuition rigorous (especially when compared against the assertion that the Euclidean ball is the “roundest” possible convex body, for which we have many rigorous and useful formulations).

To state the conjecture I need a little notation. Suppose we have a symmetric convex body $B \subset {\Bbb R}^d$ in a Euclidean space, thus B is open, convex, bounded, and symmetric around the origin. We can define the polar body $B^\circ \subset {\Bbb R}^d$ by

$B^\circ := \{ \xi \in {\Bbb R}^d: x \cdot \xi < 1 \hbox{ for all } x \in B \}$.

This is another symmetric convex body. One can interpret B as the unit ball of a Banach space norm on ${\Bbb R}^d$, in which case $B^\circ$ is simply the unit ball of the dual norm. The Mahler volume $M(B)$ of the body is defined as the product of the volumes of B and its polar body:

$M(B) := \hbox{vol}(B) \hbox{vol}(B^\circ).$