On extreme constant width bodies in $\mathbb{R}^3$

TL;DR

This work investigates extremal elements among constant width bodies in $\mathbb{R}^3$ under Minkowski addition. It establishes two broad classes of extreme shapes: all Meissner polyhedra and all axially symmetric bodies generated by rotating a symmetric Reuleaux polygon, using a combination of boundary-surgery constructions and a support-function analysis. A unified conjecture is proposed: extremality is characterized by the behavior of principal radii of curvature encoded in the support function, mirroring Kallay’s 2D criterion. The results advance the understanding of the extreme points of the convex set of 3D constant width bodies and suggest a concrete, operating criterion for extremality in terms of the generating geometry and curvature data.

Abstract

We consider the family of constant width bodies in $\mathbb{R}^3$ which is convex under Minkowski addition. Extreme shapes cannot be expressed as a nontrivial convex combination of other constant width bodies. We show that each Meissner polyhedra is extreme. We also explain that each constant width body obtained by rotating a symmetric Reuleaux polygon about its axis of symmetry is extreme. In addition, we conjecture a general characterization of all extreme constant width shapes.

Figures (11)

• Figure 1: These are Reuleaux polygons, which are constant width curves consisting of finitely many circular arcs of radius one. It turns out that each Reuleaux polygon is extreme. In particular, each one satisfies Kallay's criterion: its radius of curvature is equal to one at circular boundary points and is equal to zero at vertex points.
• Figure 2: A Meissner tetrahedron, which is an extreme constant width shape in $\mathbb{R}^3$. This shape is designed from a Reuleaux tetrahedron by performing surgery on three edges that meet in a common vertex. See Figure \ref{['ReulTetraFig']} below.
• Figure 3: This is another type of Meissner tetrahedron, which is also extreme. It is constructed from a Reuleaux tetrahedron (Figure \ref{['ReulTetraFig']}) by performing surgery on three edges which meet in a common face of the Reuleaux tetrahedron.
• Figure 4: This figure displays a rotated Reuleaux triangle on the left and a rotated Reuleaux pentagon on the right. These are two examples of extreme axially symmetric constant width shapes in $\mathbb{R}^3$.
• Figure 5: The figure on the left is a Reuleaux polyhedron $B(X)$, where $X$ is the set of vertices of a regular tetrahedron. Note that each dual edge pair of $B(X)$ is labeled with the same color. The figure on the right is the shape obtained from $B(X)$ by performing surgery on $\partial B(X)$ near one of its dual edges. The dashed curves are circular arcs of radius one which join two vertices of $B(X)$. The surgery procedure is to replace the region of $\partial B(X)$ bounded by these arcs with the surface obtained by rotating one of the arcs into the other about the line passing through the two vertices; this surface is a piece of a spindle torus, which is described in the appendix.

Theorems & Definitions (22)

• Theorem A
• Theorem B
• Theorem C
• proof : Proof of Theorem \ref{['thmA']}
• Remark 2.1
• proof : Proof of Theorem \ref{['thmB']}
• Remark 4.1
• Lemma 5.1
• proof
• proof : Proof of Theorem \ref{['thmC']}