Superlinear separation for radiant and coradiant sets

Abstract The article studies radiant and coradiant sets of some normed space X from the point of view of separation properties between a set A⊆ X and a point x ∉ A; indeed they show striking similarities with the ones holding for convex sets and can be obtained by simply changing halfspaces (level sets of linear continuous functions), with level sets of continuous superlinear functions. In a geometric perspective one can say that radiant sets are separated by means of convex coradiant sets and coradiant sets are separated by means of convex radiant sets. The identification between the geometric and the analytic approach passes through the well-known Minkoski gauge and the study of concave continuous gauges of convex coradiant sets. The results are then applied to the study of abstract convexity with respect to the family L of continuous superlinear functions, to the characterization of evenly coradiant convex sets and to the subdifferentiability of positively homogeneous functions. Dedicated to D. Pallaschke on his 65th birthday. Keywords: Abstract convexityConcave gaugeCoradiant setsPolarityRadiant setsPositively homogeneous functionsSuperlinear separationKeywords: Mathematics Subject Classifications 2000:52A3052A07 Notes Dedicated to D. Pallaschke on his 65th birthday.