A volume formula for Reuleaux polyhedra

TL;DR

This paper derives a volume formula for Reuleaux polyhedra, a special class of ball polyhedra with coinciding centers and vertices, expanding understanding in discrete and convex geometry.

Contribution

It provides the first explicit volume formula for Reuleaux polyhedra based on their edges, building on recent work on related shapes.

Findings

Derived a new volume formula for Reuleaux polyhedra

Expressed volume in terms of edge lengths

Extends geometric understanding of ball polyhedra

Abstract

A ball polyhedron is a finite intersection of congruent balls in $\mathbb{R}^3$. These shapes arise in various contexts in discrete and convex geometry. We focus on Reuleaux polyhedra, the subclass of ball polyhedra whose centers and vertices coincide. Building on Bogosel's recent work on the volume of Meissner polyhedra, we derive a formula for the volume of Reuleaux polyhedra in terms of their edges.