Typical curvature behaviour of bodies of constant width

Imre Barany, rolf Schneider

Published 2014-04-28Version 1

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.