Rectangular convexity of convex domains of constant width
A set M in the plane is called rectangular convex (r-convex) if, for any two distinct points x, y in M, there exists a rectangle R in M that contains x and y. \textit{R. Blind}, \textit{G. Valette} and \textit{T. Zamfirescu} [ibid. 9, 317-327 (1980; Zbl 0436.52007)] proved that, if M is compact, centrally symmetric, all its extreme points lie on a circle, and it has an inner point in the plane topology, then M is r-convex. They conjecture that these are the only compact r-convex sets. To support this conjecture the author proves in this paper that a closed convex domain of constant width must be a circle if it is assumed to be r-convex.
scientific article; zbMATH DE number 4136876