Convex Hull Volumes in Hyperbolic 3-Space

Cameron MacMahon

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arXiv · 2604.27291

Convex Hull Volumes in Hyperbolic 3-Space

Cameron MacMahon

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Abstract

In this paper we provide a geometric condition satisfied by certain closed subsets of the Riemann sphere which implies that their hyperbolic convex hulls in $\mathbb{H}^3$ have infinite volume. As a corollary, we characterize continua in the Riemann sphere whose hyperbolic convex hulls have infinite volume, answering a question of Danny Calegari. Furthermore, we give a geometric characterization of planar self-similar sets whose hyperbolic convex hulls have infinite volume.

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Cameron MacMahon. 2026-04-30. Convex Hull Volumes in Hyperbolic 3-Space. https://arxiv.org/abs/2604.27291

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