In the following let us consider, without essential loss functions such that $f(0)=0$.
A function $f:\mathbb{R}\to\mathbb{R}$ is called star shaped if its epigraph is star shaped with respect to $(0,0)$, or in other words:
$$\forall\alpha\in [0,1]\forall x\in\mathbb{R}\quad f(\alpha x)\leq \alpha f(x)$$
I am searching for a condition which, added to star-shapedness ensures (or even better is equivalent to) convexity of $f$.
I know that a set is convex if and only if it is star shaped with respect to any of its points. Hence a way may be building on this fact, workning on the epigraph of $f$. But I don't really know if this would give a useful sufficient condition on $f$.
Any help would be most welcomed.
2026-04-01 18:05:49.1775066749
Under which condition a star-shaped function is convex?
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