Mahler-type volume inequality for convex bodies with tetrahedral symmetry

2025/11/19 by Aliev, Arkadiy
Mathematics · Biochemistry, Genetics and Molecular Biology · #Point processes and geometric inequalities #Geometric Analysis and Curvature Flows #Diffusion and Search Dynamics

paper · doi:10.48550/arxiv.2511.14991

Abstract

Let K be a convex body in ℝⁿ . We denote the volume of K by \vert K\vert , and the polar body of its difference body K - K by (K - K)^∘ . We provide a new proof of the well-known estimate |K||(K - K)^∘| ≥ (3)/(2) for K ⊂ ℝ² , with equality attained for a triangle. For K ⊂ ℝ³ with tetrahedral symmetry, we prove that |K| |(K - K)^∘| ≥ (2)/(3), with equality attained for a tetrahedron.

Citations

• Minimal volume product of three dimensional convex bodies invariant under certain groups of order four
• The exact bound for the reverse isodiametric problem in 3-space
• Minimal volume product of three dimensional convex bodies with various discrete symmetries
• On densities of lattice arrangements intersecting every i-dimensional\n affine subspace

Related

• Isotropic constants and Mahler volumes
• A Lower Bound for the Mahler Volume of Symmetric Convex Sets
• Equipartitions and Mahler volumes of symmetric convex bodies
• On successive minima-type inequalities for the polar of a convex body
• Minimal volume product of three dimensional convex bodies with various discrete symmetries