Conjugacy for closed convex sets

Abstract

Even though the polarity is a well defined operation for arbitrary subsets in the Euclidean n-dimensional space, the related operation of conjugacy of faces appears defined in the literature exclusively for either convex bodies containning the origin as interior point and their polar sets, or for closed convex cones. This paper extends the geometry of closed convex cones and convex bodies to unbounded convex sets (and, in a dual way, to those closed convex sets containing the origin at the boundary), not only for the sake of theoretical completeness, but also for the potential applications of this theory in the fields of Convex Programming and Semi-infinite Programming. Introducing the recession cone into the analysis we develop a general theory of conjugacy which, together with the new concept of curvature index of a convex set on a face, allows us to establish a strong result on complementary dimensions of conjugate faces which extends a well-known result on polytopes.