A simple $n$-gon is a polygon with $n$ edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass asked the following question: For $n \geq 5$ odd, what is the maximum perimeter of a simple $n$-gon contained in a Euclidean unit disk?
In 2009, Audet, Hansen and Messine answered this question, and showed that the optimal configuration is an isosceles triangle with a multiple edge, inscribed in the disk. In this note we give a shorter and simpler proof of their result, which we generalize also for hyperbolic disks, and for spherical disks of sufficiently small radii.
Peeling potatoes near-optimally in near-linear time
本文考虑如下几何优化问题：求一个包含在给定的具有$n$顶点的简单多边形$P$中的面积最大的凸多边形。我们给出了一个随机的近线性时间$(1-\varepsilon)$近似算法：在$O(n( \log^2 n + (1/\varepsilon^3) \log n + 1/\varepsilon^4))$时间，我们找到一个凸多边形包含在$P$，概率至少$2/3$，面积至少$(1-\varepsilon)$倍的最优解的面积。我们也得到了类似的结果计算$P$内的凸多边形的最大周长的变体。为了实现这些结果，我们提供了新的几何概率的结果。第一个结果是与$P$内随机均匀选择的两个点相互可见的概率和$P$内最大凸体的面积相关的界限。第二个结果是任何平面凸体$K$的周长与$K$内部均匀随机样本的凸包的周长之间的差的期望值的界限。
Optimal Frames for Phase Retrieval from Edge Vectors of Optimal Polygons
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；East China Normal University（华东师范大学）
