Is there any characterization of strictly convex functions in terms of convex function, strongly convex function, or the epigraph?

Question

The definition of strict convexity is given in terms of the Jensen's inequality.

However, I am seeking alternative definition of strict convexity, either

1. stated in terms of the epigraph. For example, a function is convex if its epigraph is convex, by analogy, a function is strictly convex if its epigraph is ??

2. in terms of convex, or strongly convex function. For example, a function is strongly convex if $f(x) - \frac{1}{2}\|x\|_2^2$ is convex, by analogy, a function is strictly convex if ?? is convex/stronglyconvex.

Can we fill in the blank for ?? ?

If not, does there exist any other alternative characterization of strictly convex functions?

Answer 1

As far as I remember, the answer to 1. is also "strictly convex" (meaning that for every point of the boundary on the set there is a hyperplane that has the whole set (except this one point) on one side).

For 2. there is no answer I know of. (The function $x\mapsto \sqrt{x^2+1}$ is strictly convex, but somehow becomes less so for larger $x$.)