Mixed affine surface area
The author introduces the mixed affine area of convex bodies \(K_ 1,...,K_ n\) in \(E^ n\) having continuous curvature functions \(f(K_ i,\cdot)\) \((i=1,...,n)\) in the sense of Minkowski by means of \[ \Omega (K_ 1,...,K_ n)=\int_{S^{n-1}}[f(K_ 1,u)...f(K_ n,u)]^{1/(n+1)}dS(u), \] where \(S^{n-1}\) is the unit sphere and dS its area element. It the boundary of K is of class \(C^ 2\), then f(K,\(\cdot)\) is the reciprocal Gauss curvature as a function of the exterior unit normal vector, hence \(\Omega\) (K,...,K) is the classical affine area of K. In this paper, the mixed affine area is related to various other functionals of convex bodies, and known inequalities are combined to yield several new inequalities involving the mixed affine area. Example: \[ \Omega (K_ 1,...,K_ n)^{n+1}\leq n^{n+1} \omega^ 2_ n V(K_ 1,...,K_ n)^{n-1} \] (\(\omega\) \({}_ n=volume\) of the unit ball in \(E^ n)\) with equality if and only if the \(K_ i\) are homothetic ellipsoids.
scientific article; zbMATH DE number 4028288
scientific article