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$f: \mathbb{R}^n \rightarrow \mathbb{R},f$ is continious. $$f \text{ is convex} \Leftrightarrow f\left(\dfrac{x+y}{2}\right) \le \dfrac{f(x) + f(y)}{2}\ \ \ \forall x, y \in \mathbb{R}^n$$

One side comes straight from Jensen's inequality, while the other one is not obvious at all.

Maybe more divisions by 2 will help as I have heard.

MathTiler1
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1 Answers1

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Hint: prove by induction on $n$ that $f(tx + (1-t) y) \le t f(x) + (1-t) f(y)$ if $t = k/2^n$, $0 \le k \le n$. Then use continuity of $f$ to finish.

Robert Israel
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