Rectangular convexity of convex domains of constant width

✍ Scribed by K. Böröczky

• Publisher: Springer
• Year: 1990
• Tongue: English
• Weight: 232 KB
• Volume: 34
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00150684

✦ Synopsis

At the 1974 meeting about convexity in Oberwolfach, T. Zamfirescu raised the following problem:

Let F be a class of sets in R n. We say that a set M c R ~ is F-convex if, for any two distinct points x, y ~ M, there exists F e F such that x, y ~ F and F c M. Study the F-convexity for interesting classes F.

From this point of view, convex sets are F-convex sets when F consists of the closed segments. If the members of F are arcs, then F-convexity becomes the usual arcwise connectedness. In [1], R. Blind et al. dealt with the case when F is the class of all non-degenerate rectangles in R ~. This particular F-convexity is called rectangular convexity or, shorter, r-convexity. An rconvex set is obviously convex. For example, an open set in R * is already r-convex if it is convex. Throughout this article, we are interested only in the closed r-convex sets of R 2.

The authors of [1] described the unbounded closed r-convex sets. They also proved that, if a compact set K is centrally symmetric, all its extreme points lie on a circle, and it has an inner point in the plane topology, then K is r-convex. They conjectured that these are the only compact r-convex sets. The authors supported the conjecture with some results; however, some cases remain open, for example, the case of convex domains of constant width. In this paper we shall prove the following.