On sets of constant width

Abstract

Sets of constant width in En (En will denote Euclidean n-space) form an important subset of the set of all convex sets in En. For example, the Hausdorff k measure (0<k <n) of a subset of En is usually defined in terms of coverings of this set by arbitrary sets. However, because of the property that an arbitrary set is contained in a set of constant width of the same diameter, it is possible to refine this definition so that we need only consider coverings by sets of constant width. Unfortunately one cannot further refine the definition of Hausdorff measure, say to coverings by spheres, since it is not true that an arbitrary set is contained in a sphere of the same diameter. Thus in computing Hausdorff measures, the problem arises to find properties of sets of constant width that are analogous to those of spheres. It is the purpose of this paper to prove such a property; namely we establish a uniformity condition on sets of constant width (which incidentally does not hold for arbitrary convex sets). Roughly speaking, this uniformity condition states that parallel crosssections of a set of constant width vary continuously and that the variation for sets of given diameter d can be characterized independent of the particular set. Though the motivation given above for theorems of this type arises from measure theory, I think that the results are interesting from a purely geometric point of view. For the properties of convex sets and, in particular, sets of constant width, that will be used in this paper, the reader is referred to Theorie der konvexen Korper by J. Bonnesen and W. Fenchel, Ergebnisse der Mathematik, vol. 3, part 1, 1934. We shall in addition use the following notation: As usual, the symbol -e will denote implication between sen-