Concave Convex-Body Functionals and Higher Order Moments-of-Inertia

A class < of convex bodies (c-bodies) P,Q, . . . in a k-dimensional Euclidean space R is called convex, when for all P,Q ∈ < always follows αP × βQ ∈ < [α, β ≥ 0, α+ β = 1]. Here we interpret λP (with λ > 0) to be a c-body, obtained from P through a dilatation with respect to a fixed originO of the spaceR; P ×Q refers to Minkowski addition. This property of a class of c-bodies thus not only refers to the size and shape of the body, but also to its location in space.1 A functional φ(P ) defined over a convex class of c-bodies < is called (in the Minkowski sense) concave, when for two arbitrary (non-empty) c-bodies P,Q ∈ < the functional inequality