# Strictly convex surfaces in 3D have an equicurved point

The two principal radii of curvature at a point on a surface in three dimensions provide a succinct description of the local geometry of the surface. The principal curvatures determine whether a surface is locally flat, convex, or a saddle (Figure 1). The radii (plural of radius, pronounced ray-dee-eye or ˈrādēˌī) prove even more dispositive in the theory of convex surfaces. Convex surfaces necessarily have positive radii of curvature everywhere, but much more is true: Knowing that the two radii of curvature at every point on a convex surface in fact determines the entire surface up to translations. But when does the reverse hold?  Continue reading

# 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.