On the semicontinuity of curvatures
For a convex body \(K\) in \(\mathbb{R}^ n\) with a boundary of class \(C^ 2\) and positive curvatures and for \(\alpha\in\mathbb{R}\backslash\{0\}\) and \(j=1,\ldots,n-1\), let \[ \psi^ \alpha_ j(K)=\left\{{1\over\omega_ n}\int_{S^{n-1}}s_ j(K,u)^ \alpha d\omega(u)\right\}^{1/\alpha}, \] where \(s_ j(K,u)\) is the normalized \(j\)-th elementary symmetric function of the principal radii of curvature of \(\partial K\) at the point with outer unit normal vector \(u\); \(\omega\) is spherical Lebesgue measure on the unit sphere \(S^{n-1}\) and \(\omega_ n=\omega(S^{n-1})\). For \(\alpha=-\infty\), 0, or \(\infty\) define \(\psi^ \alpha_ j(K)=\lim_{r\to\alpha}\psi^ r_ j(K)\). The author proves that the functional \(\psi^ \alpha_ j\) can be extended to the space of all convex bodies in \(\mathbb{R}^ n\), equipped with the Hausdorff metric, in such a way that the extension is upper semicontinuous when \(-\infty\leq\alpha\leq 1\) and lower semicontinuous when \(1\leq\alpha\leq\infty\). This result generalizes earlier special cases and proves a conjecture of Wm. J. Firey.
scientific article; zbMATH DE number 167714