An Application of Shadow Systems to Mahler’s Conjecture

Autor: Artem Zvavitch, Mathieu Meyer, Matthieu Fradelizi
Přispěvatelé: Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematical Sciences Kent State University, Kent State University, U.S.~National Science Foundation grant DMS-1101636 and Université Paris-Est Marne-la-Vallée, Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)
Rok vydání: 2012
Předmět:
Zdroj: Discrete and Computational Geometry
Discrete and Computational Geometry, Springer Verlag, 2012, 48 (3), pp.721-734. ⟨10.1007/s00454-012-9435-3⟩
ISSN: 1432-0444
0179-5376
DOI: 10.1007/s00454-012-9435-3
Popis: International audience; 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.
Databáze: OpenAIRE
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