Polarity and separation of cones

Given a closed convex cone C⊂Rn and its polar cone C∘, properties of the set C∩(−C∘) are studied. In particular, we solve a problem of Stoker concerning nonemptiness of rintC∩(−rintC∘). Based on these properties, new results on separation of C and C∘ by hyperplanes are established.