A characterization of the Euclidean ball in terms of concurrent sections of constant width (Q2638528)

The author's main result reads as follows: Let \(2\leq k<n\), \(K\subset E^ n\) be an n-dimensional convex body and let \(p_ 0\) be a point of \(E^ n\) with the property that every k-section of K through \(p_ 0\) is a convex body of constant width. Then K is a Euclidean n-ball. In the case \(n=3\) this has been proved by \textit{K. Süss} [Jahresber. Dtsch. Math.- Verein. 34, 245-247 (1926)] under suitable differentiability conditions, if \(p_ 0\in int(K)\), and by the reviewer [Acta Math. Acad. Sci. Hung. 24, 135-138 (1973; Zbl 0257.53002)] without this restriction on \(p_ 0\). The proof is rather complicated and based on a detailed analysis of the location of the binormals of \(\partial K\). Additionally the author proves that every k-dimensional h-convex body, \(1<k<n\), is the k-section of an n-dimensional body of constant width h. Here a convex body K is called h-convex if for every \(p\in \partial K\) there is a supporting sphere of K through p having diameter h.