Introduce convex function on real numbers

[yoshihiro503]

Co-authored-by: Reynald Affeldt [email protected]

Motivation for this change

Things done/to do

• added corresponding entries in CHANGELOG_UNRELEASED.md
  (do not edit former entries, only append new ones, be careful:
  merge and rebase have a tendency to mess up CHANGELOG_UNRELEASED.md)
• added corresponding documentation in the headers

Automatic note to reviewers

Read this Checklist and put a milestone if possible.

[zstone1]

Nice results. I've made comments about a couple generalizations, one regarding annoying endpoint issues, the other regarding generalizing to vector space ranges. I care more about the endpoints one mostly because it's much harder to fix later. Happy to work with you on that if that'd be helpful.

[zstone1]

This definition misbehaves at the endpoints. In particular, it requires f to be be "differentiable from the left at a". That makes this definition rely on values of f outside [a,b]. One big consequence is characteristic functions like if x \in `[a,b] then 1 else 0 fails this definition, as its not derivable at a or b. For what it's worth, I had an analogous issue with realfun.v, see #752 for how I addressed it there. You'll need something slightly different here, though.

I've seen textbooks take these approaches:

1. Demanding "continuity up the endpoints" is one approach

forall x, x \in `]a, b[ -> derivable f x 1.
/\
cvg (at_right a f^`()) /\ cvg (at_right b f^`())

or some variation of this. However, this implies some continuity of the derivative, which is maybe not ideal.
2. We could change the derivative to use a different topology. One way to do that is add an analog to the {within `[a,b], continuous f} notation of {within `[a,b], derivable f}. This would take limits only on the subspace topology.
3. Similarly, but specialized to working in R, it should be fine to define it like

forall x, x \in `]a, b[ -> derivable f x 1
/\
"differentiable from the right at b" /\ "differentiable from the left at a"

For some sensible notion of "left" and "right" derivative.

[zstone1]

Where do we need the assumption that the range of f is R? If it's painless, generalizing to a vector space would be quite helpful for me. I'll need some stuff about convexity in both R and R^n for my work on space-filling curves, so I'm happy to have these results either way.

[affeldt-aist]

I opened PR #873 and moved the comments by @zstone1 there because it looked like maintainers do not have write access (not 100% sure though). I hope you don't mind moving the conversation there.