My question is, given a function $f: \mathbb{R}^n \to \mathbb{R}$ that is at least convex, is there a formal/structured way to "extend it" into a strictly convex function?
We know we can do something similar for certain non-convex function, so that it becomes convex(using the convex envelope, so global min is preserved). Is there a class of convex function for which we could make it strictly convex?
For example, suppose we are given a convex function $f(x) = 0, -1 \leq x \leq 1, x -1 > 1, -x - 1, x < 1$, then if we some how make it into $f(x) = |x|$ or $f(x) = x^2$, then the global min is preserved and we can have strict convexity.
As another example, suppose $f$ is not convex only on line given by $x = x_o + a1, a \in \mathbb{R}^n$ and strictly convex everywhere else. We can perhaps define the function so that it is strictly convex on these lines.
Ideas?
Well, something people often do in optimization is add a simple strictly convex function, e.g. consider $f(x) + \frac{\alpha}{2}\|x\|^2$ which will be $\alpha$-strongly convex (and thus strictly convex in particular).