Duality of non-exposed faces

TL;DR

This paper explores the duality relationships between non-exposed faces of convex bodies, characterizing conjugate faces via normal cones and Galois connections, with applications to planar convex bodies.

Contribution

It introduces a new Galois connection between touching cones and faces for any convex set in finite dimensions, expanding the understanding of face duality.

Findings

Characterization of conjugate faces of non-exposed faces

Introduction of a new Galois connection for convex sets

Applications to planar convex bodies and self-dual 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.