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 areĀ 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.

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