Supporting sphere

A supporting sphere for an embedded surface at $p$ is a sphere tangent at $p$ such that the surface lies locally on one side of the sphere. Comparing second fundamental forms at the tangency controls the surface's normal curvature.

Let a nonempty compact regular surface without boundary lie in a closed Euclidean ball of radius $R$. Then it has a point $p$ at which its Gaussian curvature satisfies $K(p)\ge R^{-2}$.

Maximize the squared distance to the center $c$. Put $\rho =|X(p)-c|$, so $0<\rho \le R$. Its first derivative makes the radial vector normal to the surface; choose the unit normal $N=(X(p)-c)/\rho$. For a unit tangent vector $v$, the second derivative at this maximum givesBoth principal curvatures are therefore at most $-1/\rho$. Their product is at least $1/{\rho }^2\ge 1/R^2$. Reversing the unit normal reverses both signs and leaves the Gaussian curvature unchanged.

The no-boundary hypothesis is necessary: a flat planar closed disk lies inside a ball but has zero Gaussian curvature everywhere. Its farthest points occur on the boundary, where the interior maximum argument does not apply.

An elliptic point of a surface is a point at which the Gaussian curvature is positive, equivalently where the two principal curvatures have the same nonzero sign.