论文标题
Blaschke-Santaló的类似于台球动态的不平等现象
An analogue of the Blaschke-Santaló inequality for billiard dynamics
论文作者
论文摘要
Blaschke-santaló不平等是凸面几何形状的经典不平等，涉及凸体的体积和双重偶的体积。在这项工作中，我们调查了台球动力系统中这种不平等的类似物：我们用最短的封闭台球轨迹的长度代替了体积。我们定义了一个称为凸体K的“台球产品”的数量，该数量类似于Blaschke-Santaló不平等中研究的体积产物。在平面案例中，我们根据人体直径为台球产物提供了明确的表达。我们还研究了具有固定数量的顶点的多边形类别中的上限。
The Blaschke-Santaló inequality is a classical inequality in convex geometry concerning the volume of a convex body and that of its dual. In this work we investigate an analogue of this inequality in the context of billiard dynamical system: we replace the volume with the length of the shortest closed billiard trajectory. We define a quantity called the "billiard product" of a convex body K, which is analogous to the volume product studied in the Blaschke-Santaló inequality. In the planar case, we derive an explicit expression for the billiard product in terms of the diameter of the body. We also investigate upper bounds for this quantity in the class of polygons with a fixed number of vertices.
