There is a problem with the organization. At least one of the explanatory paragraphs in "2" of the table of contents should come before "why I find it interesting".

Definitely recommend reading a little of this first:

https://en.m.wikipedia.org/wiki/Jensen%27s_inequality

(Could just provide a link to it near the beginning of the article for reference.)

Although I usually rail against Wikipedia math articles being gleefully esoteric jargon describing concepts in the most aggressively arcane way possible as if it's a competitive sport, the convex function article is fine. People have clearly done work on it

(People might say, well isn't it obvious what convex means. My response is no, it's dealer's choice)

I had the exact same reaction and was about to rant here but I looked up the wikipedia article and to be fair to the author there is no simple easy way to explain this. The article actually does a decent job. :)

"If a line's getting steeper then its average height up til now is less than its average height will be later."

The article does it around half way through,

...the inequality the way I learned it²:

E[f(x)] ≥ f(E[x])

…if f(x) is convex