Zur Affinoberfläche konvexer Körper. (On affine surfaces of convex bodies) (Q1073348)

A compact convex body K is called ''\(\epsilon\)-smooth'', if a ball of arbitrarily fixed radius \(\epsilon\) rolls freely in the interior of K, and the class of all such K is denoted by \({\mathcal G}\). Then the following generalization of a theorem of W. Blaschke (1923) in the three- dimensional analytic case holds: If K belongs to \({\mathcal G}\), then there exists \[ \lim_{\delta \to +0} c_ n[V(K)- V(K_{[\delta]},K,...,K)]/\delta^{2/(n+1)}, \] where n is the dimension of the space, \(c_ n\) a dimension constant and \(K_{[\delta]}\) the difference of K and the union of all plane sections of K with the volume \(\delta >0\). This equiaffinely invariant value may be used as a definition of an affine surface area \(A_{aff}(K)\) for K analogous to the Minkowski definition of the euclidean surface area A(K) for K. In the case of an arbitrary K \(\inf_{\tau >0} A_{aff}(K+\tau E),\) being independent from the choice of a solid ellipsoid E, plays the same role. Some properties of \(A_{aff}(K)\) are deduced from its definition. - The proofs are based on theorems of A. D. Aleksandrov concerning the twice differentiability of a convex function almost everywhere and the notion of the ''surface area function'' for K on the unit sphere.