Erdős problem 1125

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?

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FormalConjectures/ErdosProblems/

1125.lean

Retained formal statement1 of 1

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].

FormalConjectures/ErdosProblems/1125.leanErdos1125.erdos_11251 lineExact file

1True ↔ ∀ (f : ℝ → ℝ), (∀ (x h : ℝ), h > 0 → 2 * f x ≤ f (x + h) + f (x + 2 * h)) → Monotone f

SolvedProof has a holelean4external proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.