Typical curvature behaviour of bodies of constant width

Bárány, Imre and Schneider, R. (2015) Typical curvature behaviour of bodies of constant width. ADVANCES IN MATHEMATICS, 272. pp. 308-329. ISSN 0001-8708

Preview

Text 1404.7019v1.pdf Download (174kB) | Preview

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. © 2014 Elsevier Inc.

Actions (login required)

Edit Item