A Lower Bound for the Mahler Volume of Symmetric Convex Sets

2015/01/09 by Yashar Memarian, Memarian, Yashar
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Limits and Structures in Graph Theory #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1501.02009

openalex publication_date 2015/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The goal of this paper is to present a lower bound for the Mahler volume of at least 4-dimensional symmetric convex bodies. We define a computable dimension dependent constant through a 2-dimensional variational (max-min) procedure and demonstrate that the Mahler volume of every (at least 4-dimensional) symmetric convex body is greater than a (simple) function of this constant. Similar to the proof of Gromov's Waist of the Sphere Theorem in [18], our result is proved via localisation-type arguments obtained from a suitable measurable partition (or partitions) of the canonical sphere.

Related

• Equipartitions and Mahler volumes of symmetric convex bodies
• Isotropic constants and Mahler volumes
• Mahler-type volume inequality for convex bodies with tetrahedral symmetry
• Minimal volume product of three dimensional convex bodies with various discrete symmetries
• Minimal volume product of three dimensional convex bodies invariant under certain groups of order four