AI 中文总结
研究高维 Funk 球中端点势能导致的距离不对称性，针对欧几里得 beta 型径向测度确定三相极限，分别产生有向高斯金字塔、高斯-卡方 Funk 水平圆金字塔及通用有向星形金字塔，并证明末相对数贡献精确抵消。
AI 中文摘要
在高维 Funk 球中，端点势能使前向与反向距离表现出不同行为，而对称化则抹去了在金字塔极限中存续的差异。对于欧几里得 beta 型径向测度，我们确定了所有三个参数相中的自然尺度极限。发散相产生由维度与径向集中度之间的平衡所索引的有向高斯金字塔。正极限参数产生高斯-卡方 Funk 水平圆金字塔，由高斯基、卡方分布高度和有向端点势能增量构成，而消失参数则产生通用有向星形金字塔。在最后一个相中，依赖于维度的对数贡献精确抵消，因此结论无需进一步速率条件。
英文摘要
In a high-dimensional Funk ball, the endpoint potential makes the forward and reverse distances behave differently, while symmetrization erases the distinction that survives in pyramid limits. For Euclidean beta-type radial measures, we determine the natural-scale limits in all three parameter phases. The divergent phase yields directed Gaussian pyramids indexed by the balance between dimension and radial concentration. A positive limiting parameter yields a Gaussian--chi Funk horocone pyramid, built from Gaussian bases, chi-distributed heights, and a directed endpoint-potential increment, while a vanishing parameter yields the universal directed star pyramid. In the last phase, the dimension-dependent logarithmic contributions cancel exactly, so the conclusion requires no further rate condition.
Comments44 pages, 1 figure
