Hilbert Geometry Volume Asymptotics Match Hyperbolic Space Entropy — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search results are a mix of YouTube problem-solving videos (windmill problem, "hardest easy geometry," USAMO problems) and one arXiv paper on Hilbert geometry asymptotics — no single unified discovery, but the Hilbert geometry paper offers the only novel mathematical content. | MATH: Hilbert geometry volume growth entropy equals that of Lobachevsky space; asymptotic ratio of metric ball volume to metric sphere area converges to the same constant as in hyperbolic space (related to curvature −1). No explicit constants given in the abstract, but the entropy equality implies the exponential growth rate is the same as in constant negative curvature — i.e., the volume of a ball of radius R grows like e^{(n−1)R} (up to polynomial factors), with the same leading coefficient as hyperbolic space. | CONNECTION: Hilbert geometry is a Finsler geometry whose unit ball is a convex body — when that body is an ellipsoid, it reduces to Klein's model of hyperbolic space. The entropy equality
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026hilbert,
title = {{Hilbert Geometry Volume Asymptotics Match Hyperbolic Space Entropy — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635373},
url = {https://zenodo.org/records/23635373},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hilbert Geometry Volume Asymptotics Match Hyperbolic Space Entropy — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635373
UR - https://zenodo.org/records/23635373
AB - FINDING: The search results are a mix of YouTube problem-solving videos (windmill problem, "hardest easy geometry," USAMO problems) and one arXiv paper on Hilbert geometry asymptotics — no single unified discovery, but the Hilbert geometry paper offers the only novel mathematical content. | MATH: Hilbert geometry volume growth entropy equals that of Lobachevsky space; asymptotic ratio of metric ball volume to metric sphere area converges to the same constant as in hyperbolic space (related to curvature −1). No explicit constants given in the abstract, but the entropy equality implies the exponential growth rate is the same as in constant negative curvature — i.e., the volume of a ball of radius R grows like e^{(n−1)R} (up to polynomial factors), with the same leading coefficient as hyperbolic space. | CONNECTION: Hilbert geometry is a Finsler geometry whose unit ball is a convex body — when that body is an ellipsoid, it reduces to Klein's model of hyperbolic space. The entropy equality Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -