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An application of shadow systems to Mahler's conjecture.

Abstract : 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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Submitted on : Friday, February 22, 2013 - 11:35:23 PM
Last modification on : Tuesday, October 19, 2021 - 4:07:09 PM
Long-term archiving on: : Sunday, April 2, 2017 - 4:34:59 AM

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Matthieu Fradelizi, Mathieu Meyer, Artem Zvavitch. An application of shadow systems to Mahler's conjecture.. Discrete and Computational Geometry, Springer Verlag, 2012, 48 (3), pp.721-734. ⟨10.1007/s00454-012-9435-3⟩. ⟨hal-00793779⟩

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