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Coconvex sets of finite volume Schneider, Rolf


Let $C$ be a closed convex cone in $\mathbb{R}^n$, pointed and with interior points. A set $A = C \setminus K$, where $K \subset C$ is a closed convex set, is called a $C$-coconvex set if it has finite volume. The family of $C$-coconvex sets is closed under the addition $\oplus$ defined by $$C \setminus (A_1 \oplus A_2) = (C \setminus A_1) + (C \setminus A_2).$$ For compact $C$-coconvex sets, Khovanskii and Timorin (2014) have proved counterparts to the inequalities of the classical Brunn--Minkowski theory. For coconvex sets of finte volume, we prove a Brunn--Minkowski type inequality with equality discussion and then, as far as possible, counterparts to the uniqueness and existence theorems for sets with given surface area measures or cone-volume measures.

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