Erdős Problem 1125

References:

• erdosproblems.com/1125
• [Er81b] Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35.
• [Ke69] Kemperman, J. H. B., On the regularity of generalized convex functions. Trans. Amer. Math. Soc. (1969), 69-93.
• [La84] Laczkovich, M., On Kemperman's inequality $2f(x)\leq f(x+h)+f(x+2h)$. Colloq. Math. (1984), 109-115.

theorem Erdos1125.erdos_1125 : True ↔ ∀ (f : ℝ → ℝ), (∀ (x h : ℝ), h > 0 → 2 * f x ≤ f (x + h) + f (x + 2 * h)) → Monotone f

Let $f:\mathbb{R}\to \mathbb{R}$ be such that [2f(x) \leq f(x+h)+f(x+2h)] for every $x\in \mathbb{R}$ and $h>0$. Must $f$ be monotonic?

A problem of Kemperman [Ke69], who proved it is true if $f$ is measurable. Erdős [Er81b] wrote 'if it were my problem I would offer $500 for it'. This was solved by Laczkovich [La84].