三角形弦幂积分 I_2 的闭式表达式
度量几何
2026-05-27 v2 经典分析与常微分方程
概率论
摘要
弦幂积分 是平凸域在平面上由积分几何函数，通过对 chord 长度的幂次积分，针对空间中穿过该域的线的运动测度实现。我们在任意三角形上建立了 的单表达式闭式解，涉及对边长的对称对数，推导出两个解析后果：幂级数表示和锐利的等温性不等式，其显式常数包含 ，唯一地由等边三角形取得。集合 可唯一确定一个三角形的全等性，这补充了 J. Gates 通过 实现的代数识别，最小索引集为 。
引用
@article{arxiv.2605.24616,
title = {A Closed Form for the Chord-Power Integral I_2 of a Triangle},
author = {Mher Martirosyan and Ruben Sargsyan},
journal= {arXiv preprint arXiv:2605.24616},
year = {2026}
}
备注
Sections 2 and 3 of this paper (the closed-form expression for I_2, of a triangle, the power-sum series representation, and the sharp inequality) were obtained earlier by Lothar Heinrich, "On Lower Bounds of Second-Order Chord Power Integrals of Convex Discs," Preprint 27/2009, Universit\"at Augsburg. Section 4 (the recognition theorem for triangles from {I_0, I_1, I_2}) is not in Heinrich