发表机构
Aurel Vlaicu University of Arad; Université Savoie Mont Blanc; Politecnico di Milano（阿拉德奥雷尔·弗莱丘大学; 萨瓦蒙布朗大学; 米兰理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Pólya问题在多边形情形下的最优形状，证明四边形情形下对称性破缺，最优四边形为特定等腰梯形而非正方形，并用解析与区间算术方法证明。
AI 中文摘要
我们考虑Pólya问题，即在给定面积的凸集中寻找具有最长最短栅栏的形状，并限定在多边形情形，即竞争者类别被限制为具有给定边数的多边形。虽然很容易证明在三角形中，最优形状是等边三角形，但我们证明在四边形情形下会发生对称性破缺：最优四边形不是正方形。更精确地，我们将其确定为一个特定的等腰梯形，该梯形在相似变换和刚体运动下由关于其底角的初等方程唯一确定。证明结合了解析论证和严格的区间算术计算。
英文摘要
We consider Pólya's problem of finding, among convex sets of prescribed area, the one with the longest shortest fence, in the polygonal setting, namely when the class of competitors is restricted to polygons with a prescribed number of sides. While it is straightforward to show that, among triangles, the optimal shape is the equilateral one, we prove that symmetry breaking occurs in the case of quadrilaterals: the optimal quadrilateral is not the square. More precisely, we identify it as a specific isosceles trapezium, which is uniquely determined, up to homotheties and rigid motions, by an elementary equation for its base angle. The proof combines analytical arguments and rigorous interval-arithmetic computations.
