We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
An Application of Shadow Systems to Mahler's Conjecture.
- Authors
Fradelizi, Matthieu; Meyer, Mathieu; Zvavitch, Artem
- Abstract
We elaborate on the use of shadow systems to prove a particular case of the conjectured lower bound of the volume product $\mathcal{P}(K)=\min_{z\in\operatorname{int}(K)}|K||K^{z}|$, where K⊂ℝ is a convex body and K={ y∈ℝ:( y− z)⋅( x− z)≤1 for all x∈ K} is the polar body of K with respect to the center of polarity z. In particular, we show that if K⊂ℝ is the convex hull of two 2-dimensional convex bodies, then $\mathcal{P}(K) \geq \mathcal{P}(\varDelta ^{3})$, where Δ 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⊂ℝ is symmetric and the convex hull of two 2-dimensional convex bodies, then $\mathcal{P}(K) \geq \mathcal{P}(B_{\infty}^{3})$, where $B_{\infty}^{3}$ is the unit cube.
- Subjects
CONVEX bodies; CONVEX polytopes; HAUSDORFF spaces; HAUSDORFF measures; VECTOR spaces; CARATHEODORY measure
- Publication
Discrete & Computational Geometry, 2012, Vol 48, Issue 3, p721
- ISSN
0179-5376
- Publication type
Article
- DOI
10.1007/s00454-012-9435-3