Studying umbilical points on a surface, I wonder if for a convex surface there will be some similar formula to the four vertex theorem, ie will exist some minimum number of umbilical points for $ S $ convex?
Caratheodory conjecture This was proven between 1940 and 1959 by diﬀerent authors (Hamburger, Bol, Klotz 1959 ) for surfaces which are strictly convex and real analytic.
The works by Hamburger, Bol, and Klotz are concerned only with the very special case where the surface is analytic. A solution to the smooth case was announced by Guilfoyle and Klingenberg in a paper posted on the arXiv in 2008. That paper has been revised twice since then, but does not appear to have been accepted for publication yet. Some other references related to Caratheodory's conjecture may be found here.