[Paper Review] On the theory of coconvex bodies

Askold Khovanskiĭ, Vladlen Timorin|arXiv (Cornell University)|Aug 8, 2013

Point processes and geometric inequalities8 references4 citations

TL;DR

This paper extends fundamental results from convex geometry—specifically the Aleksandrov–Fenchel inequality and Ehrhart duality—to coconvex bodies, which arise as complements of convex sets within convex cones. By leveraging virtual polytopes and duality in the group of virtual convex bodies, the authors establish a coconvex version of the Aleksandrov–Fenchel inequality, demonstrating that it arises naturally from the classical convex case via algebraic relations and orthogonality conditions in the dual space.

ABSTRACT

If the complement of a closed convex set in a closed convex cone is bounded, then this complement minus the apex of the cone is called a coconvex set. Coconvex sets appear in singularity theory (they are closely related to Newton diagrams) and in commutative algebra. Such invariants of coconvex sets as volumes, mixed volumes, number of integer points, etc., play an important role. This paper aims at extending various results from the theory of convex bodies to the coconvex setting. These include the Aleksandrov-Fenchel inequality and the Ehrhart duality.

Motivation & Objective

• To generalize classical results in convex geometry—particularly the Aleksandrov–Fenchel inequality and Ehrhart duality—to the coconvex setting.
• To establish a conceptual bridge between convex geometry and singularity theory by formalizing coconvex bodies as complements of convex sets in convex cones.
• To demonstrate that invariants such as mixed volumes and integer point counts in coconvex sets are manifestations of global convex-geometric invariants.
• To show that the coconvex Aleksandrov–Fenchel inequality follows from the classical convex case via algebraic duality and orthogonality in the group of virtual polytopes.
• To formalize the analogy between coconvex geometry and algebraic geometry, particularly in terms of duality and intersection theory.

Proposed method

• The authors define coconvex bodies as the complement of a closed convex set within a closed convex cone, excluding the apex of the cone.
• They introduce the group of virtual convex bodies and use the convolution operation * to relate indicators of coconvex sets to those of convex sets via the formula $-\mathbb{I}_{A^{(i)}} = \mathbb{I}_{\Delta^{(i)}_{t_0}} * \mathbb{I}_{C_{t_0}}^{-1}$.
• Using the additivity of volume and the structure of mixed volumes, they derive the coconvex Aleksandrov–Fenchel inequality by reducing it to the classical convex case through orthogonality conditions.
• They apply the Hodge index theorem and intersection theory analogies from algebraic geometry to justify the non-positivity of certain mixed volume forms in the coconvex setting.
• The proof relies on the fact that the virtual polytope $-\mathbb{I}_{A^{(1)}}$ is orthogonal to $\mathbb{I}_{C_{t_0}}$, which implies $B(-\mathbb{I}_{A^{(1)}}, -\mathbb{I}_{A^{(1)}}) \leq 0$, leading to the coconvex inequality.
• They formalize an analogy between coconvex geometry and toric varieties, where sub-exceptional and off-exceptional divisors correspond to coconvex and cone-related objects, respectively.

Experimental results

Research questions

• RQ1 Can the classical Aleksandrov–Fenchel inequality be extended to coconvex bodies, given their role in singularity theory and local algebraic geometry?
• RQ2 How do mixed volumes and integer point counts in coconvex sets relate to global invariants in convex geometry?
• RQ3 What algebraic structure underlies the duality between coconvex and convex invariants, and how can it be formalized using virtual polytopes?
• RQ4 Is there a geometric or algebraic reason why the coconvex Aleksandrov–Fenchel form inherits the sign properties of the convex case?
• RQ5 Can the Ehrhart duality for convex polytopes be generalized to coconvex bodies using the same framework of virtual polytopes and duality?

Key findings

• The coconvex Aleksandrov–Fenchel inequality holds, with the mixed volume of $d$ coconvex bodies being non-positive under orthogonality conditions, mirroring the convex case.
• The inequality is derived from the classical convex Aleksandrov–Fenchel inequality by showing that the coconvex form is equivalent to a convex form evaluated at virtual polytopes.
• The virtual polytope $-\mathbb{I}_{A^{(1)}}$ is orthogonal to $\mathbb{I}_{C_{t_0}}$, which implies $B(-\mathbb{I}_{A^{(1)}}, -\mathbb{I}_{A^{(1)}}) \leq 0$, thus proving the coconvex inequality.
• The Ehrhart duality for coconvex bodies is established via the duality of mixed volumes and the group structure of virtual convex bodies.
• The number of integer points in coconvex bodies and their volumes are shown to be governed by the same principles as in convex geometry, with global invariants reducing local invariants in singularity theory.
• The analogy with algebraic geometry—particularly toric varieties and divisor classes—provides a conceptual framework for understanding coconvex invariants as arising from intersection theory.