What is upper convex envelope of a function

Question

Only the definition of lower convex envelope I can find in Wiki. Following the definition of lower convex envelope, I guess the definition of upper convex envelope is $$ \hat f(x) = \sup\{ g(x): g \text{ is convex and } g\ge f\} $$ whether it is right? And this definition is not visualized for me. May I have a picture about the upper convex envelope? Thanks for any help.

Answer 1

This is not based on any resources, but simply by logical reasoning, I'd say that $$\hat f\left(x\right) = \sup\left\{g\left(x\right) \middle| \text{$g$ is convex and $g \geq f$}\right\}$$ is not a reasonable definition for the upper convex envelope of $f$, since this would be solved by $\hat f\left(x\right) = \infty$ for pretty much any function $f$.

Instead, it would be reasonable to define $$\hat f\left(x\right) = \inf\left\{g\left(x\right) \middle| \text{$g$ is convex and $g \geq f$}\right\}$$ however, in contrast to the (lower) convex envelope of a function, I think that the upper convex envelope is not unique, which is why it is rarely used/mentioned. A somewhat more canonical name for $\hat f$ is minimal convex upper bound of $f$ (as this name does not imply that it is unique).

Consider for example the function $f\left(x\right) = -\exp\left(-\left|x\right|\right)^2$ on the interval $x \in \left[-1,+1\right]$ below: (blue)

The orange curve corresponds to its (lower) convex envelope and it is evident that it is unique. The two gray curves are examples of minimal convex upper bounds. It is also evident, that these curves are not unique.