A Brunn–Minkowski theory for coconvex sets of finite volume

Abstract

Let C be a closed convex cone in Rn, pointed and with interior points. We consider sets of the form A=C∖K, where K⊂C is a closed convex set. If A has finite volume (Lebesgue measure), then A is called a C-coconvex set, and K is called C-close. The family of C-coconvex sets is closed under the addition ⊕ defined by C∖(A1⊕A2)=(C∖A1)+(C∖A2). We develop first steps of a Brunn–Minkowski theory for C-coconvex sets, which relates this addition to the notion of volume. In particular, we establish the equality condition for a Brunn–Minkowski type inequality (with reversed inequality sign) and introduce mixed volumes and their integral representations. For C-close sets, surface area measures and cone-volume measures can be defined, in analogy to similar notions for convex bodies. They are Borel measures on the intersection of the unit sphere with the interior of the polar cone of C. We prove a Minkowski-type uniqueness theorem for C-close sets with equal surface area measures. Concerning Minkowski-type existence problems, we give conditions for a Borel measure to be either the surface area measure or the cone-volume measure of a C-close set. These conditions are sufficient in the first case, and necessary and sufficient in the second case.