A note on consequences of Mahler's inequalities

Yanlu Lian; Fei Xue

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arXiv · 2610.34396

A note on consequences of Mahler's inequalities

Yanlu Lian·Fei Xue

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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.

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Yanlu Lian, Fei Xue. 2026-10-25. A note on consequences of Mahler's inequalities. https://arxiv.org/abs/2610.34396

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