An isoperimetric problem for tetrahedra

It is proved that a regular tetrahedron has the maximal possible surface area among all tetrahedra having surface with unit geodesic diameter. An independent proof of O’Rourke-Schevon’s theorem about polar points on a convex polyhedron is given. A. D. Aleksandrov’s general problem on the area of a convex surface with fixed geodesic diameter is discussed. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 28–55.