In- and circumcenters of manifolds of constant width (Q756132)

A set C in a Riemannian manifold M is called definitely convex if (i) C is compact and each two points of C can be connected in C by a rectifiable curve; (ii) any shortest path in C between points of C is a geodesic; (iii) any geodesic segment in C is the unique shortest path in C between its ends; (iv) each geodesic segment in C contains no pair of conjugate points along this segment. Such a set is called convex body of constant width \(W>0\), if for any \(p\in \partial C\) and any exterior unit normal n of \(\partial C\) at p, there exists the geodesic pq of length W having direction -n at p, such that pq\(\subset C\) and pq cannot be extended in C. The author deals with in- and circumcenters of a convex body C of constant width W and its in- and circumradii \(r_ i\) and \(r_ c\). He proves that each circumcenter is an incenter and vice versa. Under certain conditions their uniqueness is established. Furthermore the equation \(r_ i-r_ c=W\) is proved which is well-known from \({\mathbb{R}}^ n\).