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