A Brunn-Minkowski theory for coconvex sets of finite volume

Abstract

Let $C$ be a closed convex cone in ${\mathbb R}^n$, pointed and with interior points. We consider sets of the form $A=C\setminus A^\bullet$, where $A^\bullet\subset C$ is a closed convex set. If $A$ has finite volume (Lebesgue measure), then $A$ is called a $C$-coconvex set. The family of $C$-coconvex sets is closed under the addition $\oplus$ defined by $C\setminus(A_1\oplus A_2)= (C\setminus A_1)+(C\setminus A_2)$. 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 conditions for a Brunn--Minkowski type inequality (with reversed inequality sign), introduce mixed volumes and their integral representations, and prove a Minkowski-type uniqueness theorem for $C$-coconvex sets with equal surface area measures.