# A Brunn-Minkowski theory for coconvex sets of finite volume

## TL;DR

This paper develops a new Brunn-Minkowski theory for coconvex sets within a convex cone, establishing inequalities, mixed volumes, and uniqueness theorems that extend classical convex geometry concepts.

## Contribution

It introduces a novel Brunn-Minkowski framework for $C$-coconvex sets, including inequalities, mixed volumes, and a Minkowski-type uniqueness theorem.

## Key findings

- Established a reversed Brunn-Minkowski inequality for $C$-coconvex sets.
- Defined mixed volumes and provided their integral representations.
- Proved a Minkowski-type uniqueness theorem for surface area measures.

## 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.