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BIRS Workshop Lecture Videos

Polytopes of Maximal Volume Product Alexander, Matt


We will discuss the maximal values of the volume product $\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K|||K^z|$, among convex polytopes $K\subset {\mathbb R}^n$ with a bounded number of vertices where $K^z = \{y\in{\mathbb R}^n : (y-z) \cdot(x-z)\le 1, \mbox{\ for all\ } x\in K\}$ is the polar body of $K$ with respect to the center of polarity $z$. In particular, we will discuss polytopes with $n+2$ vertices in $\RR^n$, symmetric polytopes with $8$ vertices in $\mathbb{R}^3$, and that the supremum is reached at a simplicial polytope with exactly $m$ vertices for all convex bodies of $m$ or fewer vertices.

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