There are several characterizations of convex functions with the Lipschitz continuous gradient. If we already know that the function is of class $C^1$, then we have the following equivalent conditions:
Theorem. Let $f\in C^1(\mathbb{R}^n)$ be convex and let $L>0$. Then the following conditions (required to hold for all $x,y\in\mathbb{R}^n$) are equivalent
- $|\nabla f(x)-\nabla f(y)|\leq L|x-y|$.
- $g(x)=\frac{L}{2}|x|^2-f(x)$ is convex.
- $f(y)\leq f(x)+\langle\nabla f(x),y-x\rangle+\frac{L}{2}|y-x|^2$.
- $\langle \nabla f(x)-\nabla f(y),x-y\rangle\leq L|x-y|^2$.
- $f(y)\geq f(x)+\langle\nabla f(x),y-x\rangle +\frac{1}{2L}|\nabla f(x)-\nabla f(y)|^2$.
- $\langle\nabla f(x)-\nabla f(y),x-y\rangle\geq\frac{1}{L}|\nabla f(x)-\nabla f(y)|^2$.
There is even a characterization of convex functions with Lipschitz continuous gradient in the class of all convex functions (without assuming a priori $C^1$ regularity as in the above result):
Theorem. For a convex function $f:\mathbb{R}^n \to \mathbb{R}$ the following conditions are equivalent
- There is $L>0$ such that for all $x,h\in\mathbb{R}^n$ $$f(x+h) + f(x-h) - 2f(x) \leq L|h|^2.$$
- $f\in C^1$ and $|\nabla f(x) - \nabla f(y)| \leq L|x-y|$ for all $x,y\in\mathbb{R}^n$.
The above results are well known, but not so easy to find. However you can found them in some monographs on convex analysis.
My questions are:
Questions 1. Are there similar characterizations of $C^2$ convex functions?
I am even interested in weaker questions:
Questions 2. Are there sufficient confitions that guarantee that a convex function is of class $C^2$?
Questions 3. Are there sufficient confitions that guarantee that a convex function of class $C^{1,1}$ is of class $C^2$?
And even a weaker one:
Questions 4. Are there sufficient confitions that guarantee that a strongly convex function is of class $C^2$?
In fact, I am interested in any non-trivial results that would imply that a given convex function is of class $C^2$. I am not aware of any such results.
