A volume formula for Reuleaux polyhedra

TL;DR

This work derives a closed-form volume formula for Reuleaux polyhedra, a subclass of ball polyhedra whose centers coincide with boundary vertices. Building on Bogosel's Meissner polyhedron volume framework, the authors decompose a Reuleaux polyhedron $B(X)$ into a Meissner body $M$ plus wedge regions, and prove a Wedge Lemma that expresses the wedge volume in terms of edge-geodesic angles via a pair of auxiliary functions $f$ and $h$. Aggregating these contributions with Bogosel's relation for $M$ yields the edge-based formula $V(B(X))=rac{2\,\pi}{3}-\frac{1}{2}\sum h(\theta(e_i),\theta(e_i'))$, and the paper establishes $h>g$ to obtain a strict Blaschke-type inequality $V(B(X))<\frac{1}{2}S(B(X))-rac{\pi}{3}$. The analysis relies on detailed sliver and spindle integral computations, specialized coordinates, and a rigorous treatment of dual-edge pairs, extending Meissner-type volume results to the broader Reuleaux polyhedron setting with explicit angular data. The results provide a practical, edge-oriented method to compute volumes of constant-width, Reuleaux-type polyhedra and clarify how edge geometry governs volumetric bounds.

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.

Figures (3)

• Figure 1: On the left is a Reuleaux polyhedron $B(X)$, where $X$ is the vertices of a regular tetrahedron with side length one. Note that there are three dual edge pairs, and each pair is labeled bold with the same color. For example, the pair $(e,e')$ is labeled in gray. Also note that the endpoints of the edge $e'$ are joined by two dashed geodesics in their respective spheres. On the right is $B(X\cup e)$, which is the figure obtained by performing surgery on $B(X)$ near $e'$. In particular, this is the figure whose boundary is obtained by cutting out the region of $\partial B(X)$ bounded by the two geodesics (which contains $e'$) and rotating one geodesic into the other about the line passing through the endpoints of $e'$.
• Figure 2: These are two views of a Meissner polyhedron $M=B(X\cup e_1\cup e_2\cup e_3)$, where $X$ is the vertices of a regular tetrahedron. The dual edge pairs are $(e_1,e_1'), (e_2,e_2')$, and $(e_3,e_3')$, and $M$ is obtained from $B(X)$ by performing surgery near the edges $e_1', e_2'$, and $e_3'$. This is a constant width body in $\mathbb{R}^3$, and its volume can be computed with formula \ref{['BogoselVolForm']}.
• Figure 3: This is a diagram of a dual curve pair $(e,e')$ and associated wedge region $W(e')$ which is bounded by the three surfaces $R_b, R_c$ and $R_s$.

Theorems & Definitions (7)

• Lemma : Wedge Lemma
• Remark 3.1
• Remark 4.1
• Lemma A.1
• proof
• Proposition A.2
• proof