We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.

We investigate the regularity of semipermeable surfaces along barrier solutions without the assumption of smoothness of the right-hand side of the differential inclusion. We check what can be said if the assumptions concern not the right-hand side itself but the cones it generates. We examine also the properties of families of sets with semipermeable boundaries.

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