论文标题
平面中凸形域的关键基因座
Critical loci of convex domains in the plane
论文作者
论文摘要
令$ k $为$ \ mathbb {r}^2 $对称性的有界凸域。 $ k $的关键基因座定义为（非空的）晶格集$λ$ in $ \ mathbb {r}^2 $最小可能的covolume，以至于$λ\ cap k = \ lbrace 0 \ rbrace $。这些是数字几何形状中的经典对象；然而，所有先前已知的关键基因座示例都是有限集或有限的封闭曲线工会。在本文中，我们提供了一种新的结构，特别是提供了具有任意Hausdorff尺寸关键基因范围的域名的示例。
Let $K$ be a bounded convex domain in $\mathbb{R}^2$ symmetric about the origin. The critical locus of $K$ is defined to be the (non-empty compact) set of lattices $Λ$ in $\mathbb{R}^2$ of smallest possible covolume such that $Λ\cap K= \lbrace 0\rbrace$. These are classical objects in geometry of numbers; yet all previously known examples of critical loci were either finite sets or finite unions of closed curves. In this paper we give a new construction which, in particular, furnishes examples of domains having critical locus of arbitrary Hausdorff dimension between $0$ and $1$.
