An application of shadow systems to Mahler's conjecture.
Résumé
We elaborate on the use of shadow systems to prove a particular case of the conjectured lower bound of the volume product $\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K|||K^z|$, where $K\subset \R^n$ is a convex body and $ K^z = \{y\in\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 show that if $K\subset \R^3$ is the convex hull of two $2$-dimensional convex bodies, then $\mathcal{P}(K) \ge \mathcal{P}(\Delta^3)$, where $\Delta^3$ is a $3$-dimensional simplex, thus confirming the $3$-dimensional case of Mahler conjecture, for this class of bodies. A similar result is provided for the symmetric case, where we prove that if $K\subset \R^3$ is symmetric and the convex hull of two $2$-dimensional convex bodies, then $\mathcal{P}(K) \ge \mathcal{P}(B_\infty^3)$, where $B_\infty^3$ is the unit cube.
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