On the semicontinuity of curvature integrals (Q2747571)

It is known that the affine surface area is upper semicontinuous. This is generalized as follows: The functions which are integrated over the boundary of a convex, not necessarily smooth, body are of the form \(f(H_j)\) where \(H_j\) is an elementary symmetric function of the principal curvatures and \(f\) is a concave function vanishing at 0 such that \(f(t)/t \to 0\) for \(t\to\infty\). It is shown that such integrals are upper semicontinuous. The same is shown if the \(H_j\) are replaced by elementary symmetric functions of the principal radii of curvature. The relations between these types of integrals are discussed.