Funk Beta Balls: Exact Potentials and a Three-Phase Pyramid Diagram
Abstract
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.
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BibTeX
@misc{yokota2026funk,
title = {{Funk Beta Balls: Exact Potentials and a Three-Phase Pyramid Diagram}},
author = {Shigeaki Yokota},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27752},
eprint = {2609.27752},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27752},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Funk Beta Balls: Exact Potentials and a Three-Phase Pyramid Diagram
AU - Shigeaki Yokota
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27752
UR - https://arxiv.org/abs/2609.27752
AB - 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.
ER -