[Paper Review] Duality of non-exposed faces

Stephan Weis|arXiv (Cornell University)|Jul 12, 2011

Point processes and geometric inequalities13 references5 citations

TL;DR

This paper establishes a duality theory for non-exposed faces in convex bodies using normal cones and Galois connections between faces and touching cones. It proves that conjugate faces of non-exposed faces are singular (with incomplete normal cones), and characterizes them via mixed and free corners in planar convex bodies, providing a complete duality framework for non-exposed points in dual pairs of convex bodies.

ABSTRACT

Given any polar pair of convex bodies we study its conjugate face maps and we characterize conjugate faces of non-exposed faces in terms of normal cones. The analysis is carried out using the positive hull operator which defines lattice isomorphisms linking three Galois connections. One of them assigns conjugate faces between the convex bodies. The second and third Galois connection is defined between the touching cones and the faces of each convex body separately. While the former is well-known, we introduce the latter in this article for any convex set in any finite dimension. We demonstrate our results about conjugate faces with planar convex bodies and planar self-dual convex bodies, for which we also include constructions.

Motivation & Objective

• To characterize the duality of non-exposed faces in polar pairs of convex bodies using normal cones.
• To establish a Galois connection between touching cones and faces of convex sets, introducing a new duality framework for general finite-dimensional convex sets.
• To demonstrate that conjugate faces of non-exposed faces are singular, i.e., have normal cones of dimension at least two.
• To construct planar self-dual convex bodies without non-exposed points, using symmetric gluing of dual convex bodies.
• To apply the theory to quantum information and convex algebraic geometry, particularly in studying discontinuities in information measures and determinantal varieties.

Proposed method

• Uses the positive hull operator to define lattice isomorphisms linking three Galois connections: one between dual convex bodies, and two between touching cones and faces of each body.
• Introduces a new Galois connection between touching cones and faces of a convex set, which is general and valid in any finite dimension.
• Applies the conjugate face map to non-exposed faces, showing they correspond to mixed or free corners in the dual body.
• Employs the support function and radial function to parametrize the boundary of the dual convex body via the unit sphere.
• Uses Corollary 16.5.2 from [Ro] to compute duals of convex hulls, enabling explicit constructions of dual and self-dual bodies.
• Applies Carathéodory’s theorem to prove compactness and absence of non-exposed points in convex hulls of dense sets on the unit circle.

Experimental results

Research questions

• RQ1 How are non-exposed faces of a convex body related to the faces of its polar dual via the conjugate face map?
• RQ2 What geometric and topological properties characterize the conjugate of a non-exposed face in a dual pair of convex bodies?
• RQ3 Can planar self-dual convex bodies without non-exposed points be systematically constructed, and what conditions ensure their existence?
• RQ4 How do incomplete normal cones relate to singular faces, and how can they be described in planar convex bodies?
• RQ5 What role do mixed and free corners play in the duality of non-exposed points in two-dimensional convex sets?

Key findings

• The conjugate face map restricts to a surjection from the set of non-exposed points of a planar convex body onto the set of mixed and free corners of its polar body.
• A face conjugate to a non-exposed face is singular, meaning its normal cone has dimension at least two, and such faces are fully characterized by incomplete normal cones.
• All corners of a planar convex body without non-exposed points are polyhedral, and such bodies can be constructed via symmetric gluing of dual convex bodies along a line.
• In planar convex bodies, incomplete normal cones correspond to mixed or free corners, which are geometrically described as points where the boundary has both tangential and conical behavior.
• Self-dual convex bodies without non-exposed points exist and can be constructed by combining a convex body and its dual across a line, provided the original body has smooth extremal points on the axis of symmetry.
• Example 5.5 constructs a self-dual convex body with infinitely many corners, all polyhedral, by taking the convex hull of a dense, symmetric subset of the unit circle, ensuring no non-exposed points exist.