AI 中文总结
本研究结合计算与严格分析，证明负指数Riesz容量比的尖锐不等式，提出奇数多边形平衡测度猜想，并证明高维正单纯形优于球，附代码支持复现。
AI 中文摘要
我们通过结合计算实验与严格分析，研究了负指数Riesz容量比的尖锐不等式。对于直线上的有限子集，我们证明了当$-1<p<0$时平衡质量的正性，从而能够对猜想中的极值比进行数值检验。在平面中，将圆盘与正多边形顶点集进行比较，揭示了所测试竞争者之间的一系列转变，并为奇数多边形的平衡测度提出了一个精确猜想，我们给出了部分证明。等式曲线的数值交点表明，这些集合优于圆盘的区域并非简单嵌套。类似的数值交点也出现在三维空间中，即正单纯形等式曲线与显式五点及六点构型的等式曲线之间。受这些比较的维度依赖性的启发，我们证明了对于每个固定的$p<-2<q<0$，在所有足够大的维度中，正单纯形比球具有更大的容量比。随附的Python和Mathematica代码支持猜想的复现和进一步测试。
英文摘要
We investigate sharp inequalities for ratios of Riesz capacities with negative exponents by combining computational experiments with rigorous analysis. For finite subsets of the line, we prove positivity of equilibrium masses when $-1<p<0$, enabling numerical tests of conjectured extremal ratios. In the plane, comparisons of the disk with regular polygon vertex sets reveal a cascade of transitions among the tested competitors and suggest a precise conjecture for the equilibrium measure of odd polygons, for which we give a partial proof. Numerical intersections of equality curves show that the regions where these sets outperform the disk are not simply nested. Similar numerical intersections occur in three dimensions between the regular-simplex equality curve and those of explicit five-point and six-point configurations. Motivated by the dimensional dependence of these comparisons, we prove that for each fixed $p<-2<q<0$, the regular simplex has a larger capacity ratio than the ball in all sufficiently large dimensions. Accompanying Python and Mathematica code supports reproduction and further testing of the conjectures.
Commentsfor associate code, see https://github.com/vhdvhd/Riesz-capacity-ratios-with-negative-exponents
