New estimates for $d_{2,1}$ and $d_{3,2}$

Arkadiy Aliev

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arXiv · 2207.09552

New estimates for $d_{2,1}$ and $d_{3,2}$

Arkadiy Aliev

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Abstract

Let $K$ be a convex body in $\mathbb{R}^{n}$. Let $ d_{n,n-1}(K)$ be the smallest possible density of a non-separable lattice of translates of $K$. In this paper we prove the estimate $d_{2,1}(K)\leq \frac{\pi\sqrt{3}}{8}$ for $K\subset \mathbb{R}^{2}$, with equality if and only if $K$ is an ellipse, which was conjectured by E. Makai. Also we prove the estimate $d_{3,2}(K)\leq\frac{\pi}{4\sqrt{3}}$ for $K\subset\mathbb{R}^{3}$ using projection bodies.

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Arkadiy Aliev. 2022-07-19. New estimates for $d_{2,1}$ and $d_{3,2}$. https://arxiv.org/abs/2207.09552

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