On the geodesic diameter of surfaces possessing an involutory isometry. (Russian) Zbl 1015.53041

The main result reads as follows: Let \((M,\rho)\) be a metric space endowed with an inner metric which is homeomorphic to a round sphere \(S^2\) and let \(I\: M\to M\) be a fixed point free involutory isometry. Then there exists \(x\in M\) such that the geodesic distance between \(x\) and \(I(x)\) equals the geodesic diameter of \(M\). This implies that the geodesic diameter of the boundary of a convex centrally-symmetric body in the Euclidean 3-space is equal to the maximal geodesic distance between symmetric points.


53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
53A05 Surfaces in Euclidean and related spaces