双曲空间中的Pal等最小宽度问题
Pal's isominwidth problem in the hyperbolic space
AI总结：
本文探讨双曲空间中的Pal等最小宽度问题，发现H^n中凸体的等最小宽度问题无意义，但在H^2的水平圆凸体中，正水平圆三角形是给定Lassak宽度下的面积极小化子。
摘要：
本文聚焦于1921年R^2中经典Pal等最小宽度不等式的可能双曲版本，该不等式指出：对于固定最小宽度，正三角形面积最小。我们注意到，当n>2时，R^n中的等最小宽度问题仍未解决。近期关于球面S^2上等最小宽度问题的研究显示，根据Bezdek和Blekherman的结果，当宽度不超过π/2时，解为正球面三角形；而Freyer和Sagmeister证明，当最小宽度大于π/2时，极小化子是球面Reuleaux三角形的极点。本文讨论了双曲空间H^n中基于Lassak定义的最自然宽度概念的等最小宽度问题。一方面，我们证明H^n中具有给定最小Lassak宽度w>0的凸体体积可任意小，因此H^n中凸体的等最小宽度问题无意义；另一方面，在二维情形下，我们证明H^2中具有给定Lassak宽度的水平圆凸体中，正水平圆三角形面积最小。
英文摘要：
The paper focuses on possible hyperbolic versions of the classical Pal isominwidth inequality in R^2 from 1921, which states that for a fixed minimal width, the regular triangle has minimal area. We note that the isominwidth problem is still wide open in R^n for n>2. Recent work on the isominwidth problem on the sphere S^2 shows that the solution is the regular spherical triangle when the width is at most π/2 according to Bezdek and Blekherman, while Freyer and Sagmeister proved that the minimizer is the polar of a spherical Reuleaux triangle when the minimal width is greater than π/2. In this paper, the hyperbolic isominwidth problem is discussed with respect to the probably most natural notion of width due to Lassak in the hyperbolic space H^n where strips bounded by a supporting hyperplane and a corresponding hypersphere are considered. On the one hand, we show that the volume of a convex body of given minimal Lassak width w>0 in H^n might be arbitrarily small; therefore, the isominwidth problem for convex bodies in H^n does not make sense. On the other hand, in the two-dimensional case, we prove that among horocyclically convex bodies of given Lassak width in H^2, the area is minimized by the regular horocyclic triangle.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
