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arXiv · 1911.12874

Brunn-Minkowski type inequalities for the lattice point enumerator

David Iglesias·Jesús Yepes Nicolás·Artem Zvavitch
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Abstract

Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator $\mathrm{G}_n(\cdot)$ are provided. In particular, we show that $$\mathrm{G}_n((1-\lambda)K + \lambda L + (-1,1)^n)^{1/n}\geq (1-\lambda)\mathrm{G}_n(K)^{1/n}+\lambda\mathrm{G}_n(L)^{1/n}$$ for any non-empty bounded sets $K, L\subset\mathbb{R}^n$ and all $\lambda\in(0,1)$. We also show that these new discrete versions imply the classical results, and discuss some links with other related inequalities.

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David Iglesias, Jesús Yepes Nicolás, Artem Zvavitch. 2019-11-28. Brunn-Minkowski type inequalities for the lattice point enumerator. https://arxiv.org/abs/1911.12874

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