Typical curvature behaviour of bodies of constant width

TL;DR

This paper investigates the typical curvature behavior of convex bodies with constant width, revealing that most boundary points have curvatures equal to one, contrasting with bodies of arbitrary shape.

Contribution

It establishes that for typical convex bodies of constant width, the boundary curvatures are almost everywhere equal to one, extending known phenomena to this special class of bodies.

Findings

Most boundary points have curvature equal to one.

Contrasts with bodies of arbitrary shape where curvatures can be zero or infinite.

Uses linear curvature notions via tangential radii of curvature.

Abstract

It is known that an $n$-dimensional convex body which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical $n$-dimensional convex body of constant width $1$ (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to $1$. (In contrast, note that a ball of width $1$ has radius $1/2$, hence all its curvatures are equal to $2$.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.