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Miklós Schweitzer 2001 Problem 6

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February 12, 2017
Miklos SchweitzerfunctioninequalitiesFunctional inequalityreal analysiscollege contests

Problem Statement

Let I⊂RI\subset \mathbb R be a non-empty open interval, ε≥0\varepsilon\geq 0 and f ⁣:I→Rf\colon I\rightarrow\mathbb R a function satisfying the f(tx+(1−t)y)≤tf(x)+(1−t)f(y)+εt(1−t)∣x−y∣f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y| inequality for all x,y∈Ix,y\in I and t∈[0,1]t\in [0,1]. Prove that there exists a convex g ⁣:I→Rg\colon I\rightarrow\mathbb R function, such that the function l:=f−gl :=f-g has the ε\varepsilon-Lipschitz property, that is ∣l(x)−l(y)∣≤ε∣x−y∣ for all x,y∈I|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I
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