To prove the convexity of function $g = \inf_{\alpha > 0} \frac{f(\alpha x)}{\alpha}$.

Question

Let $f$ be a convex function. Define the function $g$ as $$ g(x) = \inf_{\alpha > 0} \dfrac{f(\alpha x)}{\alpha}. $$

Show, that $g$ is convex.

I have tried to prove it using the definition but there are no results.

Answer 1

This is easier to prove by combining two standard results from convex analysis:

If $t>0$ then the perspective of a convex $f$ is $\pi(x,t)= t f( {x \over t})$ which is convex in $(x,t)$.

If $(a,b) \mapsto w(a,b)$ is convex in $(a,b)$ and $B$ is convex, then the function $a \mapsto \inf_{b \in B} w(a,b)$ is convex.

Finally note that $g(x) = \inf_{\alpha >0} {f ( \alpha x) \over \alpha} = \inf_{t >0} t f( {x \over t}) = \inf_{t >0} \pi(x,t) $ and so is convex.