Equality of two representations of extended affine surface area
Let \(d \geq 2\) and denote by \(K^d\) the class of all nonempty compact, convex sets of Euclidean \(d\)-space \(\mathbb{R}^d\). The authors prove the Theorem: For an arbitrary convex body \(K \in K^d\) the two notions of extended equiaffine surface area due to Lutwak respectively Leichtweiß coincide, i.e. \(\Omega (K) = \widetilde \Omega (K)\) for all \(K \in K^d\). Denote by \(S^d\) the set of all star bodies in \(\mathbb{R}^d\) and by \(S^d_c (\subset S^d)\) the set of the centered star bodies in \(\mathbb{R}^d\) \((L \in S^d\) is called centered if it is a star body with respect to its centre of gravity). Using a well-defined function \(w(K,L)\) for \(K \in K^d\), \(L \in S^d\) one has \(\Omega (K) : = \inf \{w (K,L) |L \in S^d_c\} \geq \widetilde \Omega (K) : = \inf \{w(K,L) |L \in S^d\}\). Thus it is sufficient to show that \(\Omega (K) \leq \widetilde \Omega (K)\) for \(K \in K^d\). One considers a set of star bodies \(L (\varepsilon, z)\) which depend on parameters \(\varepsilon > 0\) and \(z \in \mathbb{R}^d \backslash \{\vec o\}\). The star bodies \(L\) and \(L (\varepsilon,z)\) differ only in the direction of vectors which belong to a spherical \(\varepsilon\)-neighbourhood of \(z/ |z |\). For \(\varepsilon \to 0\) the difference of the volumes of \(L\) and \(L(\varepsilon,z)\) can be made arbitrarily small. Additional considerations --- which include the homotopy invariance of the degree of a map --- finally yield the existence of \(L (\varepsilon, z (\varepsilon)) \subset S^d_c\) such that \(w(K,L (\varepsilon, z (\varepsilon))) \to w (K,L)\) for \(\varepsilon \to 0\).
scientific article; zbMATH DE number 815700