Skip to Main content Skip to Navigation
Journal articles

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.
Document type :
Journal articles
Complete list of metadatas

Cited literature [26 references]  Display  Hide  Download

https://hal-upec-upem.archives-ouvertes.fr/hal-00793779
Contributor : Matthieu Fradelizi <>
Submitted on : Friday, February 22, 2013 - 11:35:23 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM
Long-term archiving on: : Sunday, April 2, 2017 - 4:34:59 AM

File

FMZ-final.pdf
Files produced by the author(s)

Identifiers

Citation

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⟩

Share

Metrics

Record views

619

Files downloads

1021