Curvature relations and affine surface area for a general convex body and its polar
The author investigates the connection of curvature invariants for the boundary of an arbitrary convex body \(K\) in \(\mathbb{R}^d\) with the origin 0 in its interior and for the boundary of its polar body \(K^*\) with respect to 0. His main result is the formula \[ H_{d-1} (K,x_0) \cdot H_{d-1} (K^*,x^*_0) = \bigl(|x_0 |\cdot |x^*_0 |\bigr)^{-(d+1)} >0 \] for the \(({\mathcal H}^{d-1}\)-almost all) normal boundary points \(x_0\) of \(K\) resp. \(x^*_0\) of \(K^*\) with \(\langle x_0, x^*_0\rangle =1\) where \(H_{d-1}\) denotes the Gauss curvature. Another version of this formula says that the values of the equiaffine distance from the origin \((H_{d-1})^{- {1 \over d+1}} \cdot h\) \((h\) support function) of \(K\) and \(K^*\) at corresponding points are reciprocal. Moreover the author proves as an application that his so-called \(p\)-affine surface area \({\mathcal O}_p (K)\) \((p>0)\) of \(K\) [see the author, `Contributions to affine surface area', Preprint Freiburg Br. (1995)] satisfies the relation \({\mathcal O}_p (K)= {\mathcal O}_{{d^2 \over p}} (K^*)\). Since \({\mathcal O}_d (K)\) is the centroaffine surface area of \(K\) this relation generalizes the classical statement that the centroaffine surface areas of \(K\) and \(K^*\) coincide if the boundary of \(K\) is of class \({\mathcal C}^3_+\).
scientific article; zbMATH DE number 920808