I am trying to get a hold on exactly what strong convexity gives you over strict (or regular) convexity.
Yes there are simple functions which demonstrate the difference between these ideas, but what are some situations in which strong convexity makes possible some proof/algorithm which otherwise fails?
Basically: If strong convexity is assumed, how much generality is lost and how much extra regularity is gained?
(The application area I am most interested in dealing with is convex optimization, but other examples would be appreciated too)