# Convexity and positive functions

In this post, we offer a proof of another classical result: A sufficiently smooth $$1$$-homogeneous function $$h\colon \mathbb{R}^2\to \mathbb{R}$$ is convex if and only if

$h(\theta) + h”(\theta) \ge 0 \text{ for all } 0 \le \theta < 2\pi,$

where $$h(\theta):= h(\cos(\theta), \sin(\theta)$$ represents evaluation of $$h$$ on the unit circle. Since one-homogeneous convex functions are support functions, the equation above characterizes convex bodies in two dimensions.

The non-standard proof below provides a more modern route to the result than classical approaches. I expect that this technique extends to higher dimensions more directly than the classical arguments available, e.g., in Groemer’s text.