There are several equivalent definitions for strongly convex.
For example, some literature said:
A function $f$ is strongly convex with modulus $c$ if either of the following holds
-
$$f(\alpha x+(1-\alpha)x')\leq\alpha f(x)+(1-\alpha)f(x')-\frac{1}{2}c\alpha(1-\alpha)\|x-x'\|^2$$
-
$f-\frac{c}{2}\|\cdot\|^2$ is convex.
I do not know how to prove the equivalence of the above statements.
The difficulty here is that the norm is an arbitrary norm, not necessarily the $\ell_2$ norm.
Best Answer
I think you need to say that the space is an inner product space, in which case the equivalence can fairly easily be obtained. You can find the proof here.