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I'm trying to prove the following know result:

Theorem Let $I = (a,b)$ be an interval, and let $\phi \colon I \to \mathbb{R} $ be a continuos function. Then, $\phi$ is convex in $I$ if and only if $$(*) \quad \phi(x) \leq \dfrac{1}{2\delta}\int_{x-\delta}^{x+\delta}\phi(t)dt,$$ for all $a < x-\delta < x < x+\delta < b$.

It is "easy" to prove, via Jensen's inequality, that if $\phi$ is convex then it satisfies $(*)$. However, I'm stuck proving the converse.

I've read that it can be deduced from the following result:

Lemma In the upper setting, $\phi$ is convex in $I$ if and only if for any given $\alpha \in \mathbb{R}$ and any interval $J = [c,d] \subset I$ the function $\phi+\alpha x$ attains its maximum in any of the limit points of $J$, that is, $c$ or $d$.

Any advice on how to finish the proof?

Thanks in advance.

fcz
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    Looks like a duplicate of https://math.stackexchange.com/q/927824/42969 – Martin R Jan 20 '21 at 17:08
  • Both questions seem related. I'll give it a go! Thanks. – fcz Jan 20 '21 at 17:38
  • In case it helps anyone: the ideas given in the post that @MartinR suggested also help to prove the problem I posted (and, indeed, are based in the lemma I gave). Thank you, mate. – fcz Jan 20 '21 at 20:56

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