Year: 2026
Type: preprint
Abstract: We derive Makai's inequality $\vol_n(K)\vol_n((K-K)^\circ)\ge(n+1)/n!$ in every dimension from the symmetric Mahler theorem in one higher dimension, with equality precisely for simplices. The proof uses a classical two-layer construction, an identity for its polar volume, and a Rogers--Shephard volume estimate. We then record basic consequences of the Mahler and Makai inequalities for polar successive minima, successive minima at the Santaló point, critical determinants, first covering minima, and non-separable lattice arrangements. We also combine Alexandrov--Fenchel with the volume-product inequalities to obtain sharp bounds for paired mixed volumes.
Source: arXiv (Cornell University)
Authors: Lian, Yanlu, Xue, Fei
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: 若干几何体覆盖问题在格理论中的研究
OpenAlex ID: W7224021822