Supporting-sphere curvature bound
= Supporting-sphere curvature bound
If a compact regular surface lies in a closed Euclidean ball of radius $R$, maximize distance from the ball's centre. At a maximizing point, comparison of the <second fundamental form> with the tangent <supporting sphere> makes both <principal curvature>[principal curvatures] have magnitude at least $R^{-1}$ and the same sign. The <Gaussian curvature> there is therefore at least $R^{-2}$.