Surfaces of generalized constant width (Q1334388)

In the sense of the author an orientable closed connected surface \(\Phi\) in \(E^ 3\) is called a surface of generalized constant width \(d\), if \(\overrightarrow{0p} + d n(p)\) is located on \(\Phi\) for all \(p \in \Phi\) and if this map from \(\Phi\) to \(\Phi\) is an involution \(\delta\), \(n(p)\) denoting the interior unit normal of \(\Phi\) at \(p\). This implies that \(\Phi\) admits an involutive self-parallelism in the sense of \textit{H. R. Farran} and \textit{S. A. Robertson} [J. Lond. Math. Soc., II. Ser. 35, 527- 538 (1987; Zbl 0623.53022)]. The aim of this paper is to show that such an analytic surface \(\Phi\) must be a sphere of diameter \(a\), if the absolute values of the Gaussian curvatures at \(p\) and \(\delta(p)\) coincide. The proof immediately follows from Christoffel's characterization of the sphere if \(\Phi\) is assumed to be convex. In the general case the hard part of the proof is to show, that \(\Phi\) contains two focal points with common base \(p\) in the interior of the segment \(\overline{p \delta(p)}\) for some \(p\).