论文标题
$ d_ {2,1} $和$ d_ {3,2} $的新估计值
New estimates for $d_{2,1}$ and $d_{3,2}$
论文作者
论文摘要
令$ k $为$ \ mathbb {r}^{n} $中的凸形主体。令$ d_ {n，n-1}（k）$为$ k $的不可分割的晶格的最小密度。在本文中，我们证明了估计$ d_ {2,1}（k）\ leq \ frac {π\ sqrt {3}} {8} {8} {8} $ for $ k \ subset \ subset \ subset \ mathbb {r}^{2}^{2 {2 {2} $，只有$ k $是enepse，均为eneipse，均为enepeiun。另外，我们还证明了估计$ d_ {3,2}（k）\ leq \fracπ{4 \ sqrt {3}} $使用投影主体使用$ k \ subset \ subset \ subset \ mathbb {r}^{3} $。
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 $d_{2,1}(K)\leq \frac{π\sqrt{3}}{8}$ 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 $d_{3,2}(K)\leq\fracπ{4\sqrt{3}}$ for $K\subset\mathbb{R}^{3}$ using projection bodies.
