Delaunay triangulations of lens spaces

TL;DR

This paper studies the geometric structure of lens spaces by analyzing the convex hulls of finite subgroups in complex space, revealing combinatorial patterns linked to continued fractions and providing a new proof of a known theorem.

Contribution

It offers a new proof of Smilansky's theorem on convex hulls of finite subgroups in complex space, with enhanced intermediate results and combinatorial insights.

Findings

Convex hulls are characterized by continued fraction patterns.

Provides a stronger intermediate step in the proof of Smilansky's theorem.

Connects geometric structures of lens spaces with combinatorial continued fraction data.

Abstract

We compute the convex hull in $\mathbb{C}^2$ of an arbitrary finite subgroup of ${\mathbb{C}^*}^2$. The combinatorics are dictated by continued fractions in a natural way. This reproves a theorem of Smilansky, with a slightly stronger intermediary step.