Simple example where strong convexity is necessary over strict (or even regular) convexity.

Question

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)

Answer 1

Strict convexity gives uniqueness of minimizers. Strong convexity implies that the Hessian is uniformly positive definite (which suggests reasonable numerical behavior).

Answer 2

Another thing: the Legendre transform of a strictly convex function is.differentiable. The Legendre transform of a strongly convex function has Lipschitz gradient.

But frankly, your question sounds like "give a list of statements that are true for class A and are false for class B". Not likely to generate much interest.