$f(x)$: Jensen's inequality holds but not convex

Asked May 12 '22 at 17:51

Active May 12 '22 at 17:56

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It is known, that if $f(x):\mathbb{R} \to \mathbb{R}$ is a continuous function defined on $(a,b)$ such that $$f\left(\frac{x+y}{2}\right)\leq \frac{1}{2} f(x) + \frac{1}{2} f(y) \tag{*}$$ for all $x,y \in (a,b)$, then $f$ is convex.

Hence continuity is a satisfactory condition for the convexity given $(*)$.

Is there an example of a function $g$ that is not continuous and satisfies $(*)$?
Is there an example of a function $g$ that is not convex and satisfies $(*)$?

I am pretty sure such a question has been asked, yet I can't find a thing.

edited May 12 '22 at 17:56

asked May 12 '22 at 17:51

Csaba Daniel Farkaš

1

Property (*) is called “midpoint convex”. There are various Q&As about when midpoint-convexity implies convexity (e.g. if $f$ is continuous, or just bounded) – Martin R May 12 '22 at 18:09
Thanks! Knowing the name of the property helped hah! – Csaba Daniel Farkaš May 12 '22 at 18:14
Here is a quite strong result: https://math.stackexchange.com/q/4354554/42969 – Martin R May 12 '22 at 18:17
With the axiom of choice, Hamel proved that there are discontinuous functions satisfying the Cauchy equation $f(x+y)=f(x)+f(y)$ for any $x, y\in\mathbf R.$ – Rurifangtun May 13 '22 at 02:19
Refer to https://link.springer.com/article/10.1007/BF01457624 – Rurifangtun May 13 '22 at 02:22

$f(x)$: Jensen's inequality holds but not convex

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