The definition of strict convexity is given in terms of the Jensen's inequality.
However, I am seeking alternative definition of strict convexity, either
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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 ??
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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?
Best Answer
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 $x$.)