On the amount of minimal pairs of convex sets

Abstract

In this paper, we determine the cardinality of various subsets of the family of all minimal pairs of compact convex subsets of ℝ ⁿ (Theorems 2.1 and 2.8). Furthermore, we express minimality properties of a pair of closed bounded convex sets in set–theoretical terms (Theorem 3.3).

Keywords:

• convex analysis
• minimal pairs of compact convex sets

AMS Subject Classification :

• 52A07
• 26A27

Acknowledgements

The authors thank an anonymous referee for his valuable remarks on the first version of this paper and for his short proof of Lemma 2.3.