FormalConjectures.ErdosProblems.«1125»

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import FormalConjecturesUtil

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.


namespace Erdos1125

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].

@[category research solved, AMS 26, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1125.lean"]
theorem declaration uses 'sorry'erdos_1125 : answer(True) ↔ ∀ (f : ℝ → ℝ)
    (hf : ∀ x : ℝ, ∀ h : ℝ, h > 0 → 2 * f x ≤ f (x + h) + f (x + 2 * h)),
    Monotone f := by⊢ True ↔ ∀ (f : ℝ → ℝ), (∀ (x h : ℝ), h > 0 → 2 * f x ≤ f (x + h) + f (x + 2 * h)) → Monotone f
  sorryAll goals completed! 🐙


end Erdos1125