Bemerkungen zur Definition einer erweiterten Affinoberfläche von E. Lutwak. (Remarks on the definition of extended affine surface area given by E. Lutwak)
Let K denote a compact convex body in \(R^ n\). In a previous paper, the author has defined the affine area of K by the formula \[ O_{aff}(\partial K)=\lim_{\epsilon \to +0}[\lim_{\delta \to +0} nc_ n(V(K^ E_{\epsilon})-V(K^ E_{\epsilon},...,K^ E_{\epsilon},(K^ E_{\epsilon})_{[\delta]})\delta^{-2/(n+\quad 1)} \] where \(c_ n=2(\omega_{n-1}/(n+1))^{2/(n+1)}\), \(K^ E_{\epsilon}=K+\epsilon E=E\)-parallel body to K at distance \(\epsilon\) \((E=ellipsoid\) of volume \(V(E)=\omega_ n)\) and \((K^ E_{\epsilon})_{[\delta]}=residual\) body of \(K^ E_{\epsilon}\) of parameter \(\delta\) [see the author, Manuscr. Math. 56, 429-464 (1986; Zbl 0588.52011)]. Independently, E. Lutwak has defined the affine area \(\Omega\) (\(\partial K)\) of a compact convex body K in \(R^ n\) by the formula \[ (\Omega (\partial K))^{(n+1)/n}=\inf_{L\in S^ n_ 0}\{(\int_{\partial B}\rho_ L^{-1}d\quad O_ K)\quad (nV(L))^{1/n}\} \] where B denotes the unit n-sphere with the center at the origin, \(S^ n_ 0\) denotes the set of starshaped bodies L in \(R^ n\) of volume V(L) defined by the radial function \(\rho_ L(\xi)\) i.e. \[ L=\{x\in R^ n;\quad x=\alpha \xi,\quad \xi \in \partial B,\quad 0\leq \alpha \leq \rho_ L(\xi)\} \] which have the centroid at the origin. The present paper is devoted to compare with each other these two definitions. It turns out that \(O_{aff}(\partial K)\) coincides with a modified expression \({\tilde \Omega}\)(\(\partial K)\) such that \({\tilde \Omega}(\partial K)\leq \Omega (\partial K).\) For convex bodies K with continuous and positive curvature function \(\rho\), there is \[ \Omega (\partial K)=0_{aff}(\partial K)=\int_{\partial B}\rho^{n/(n+1)}d\sigma. \] Other properties of \(\Omega\) (\(\partial K)\), \(O_{aff}(\partial K)\) and their relations with the geominimal surface area G(\(\partial K)\) of C. M. Petty are also given.
scientific article; zbMATH DE number 4122759