What Convex Geometries Tell About Shattering-Extremal Systems

Abstract

We give a characterization of shattering-extremal set systems in terms of forbidden projections, in the spirit of Dietrich's characterization of antimatroids. Apart from that, we prove several metric and topological properties of such systems, which, however, do not amount to a characterization. The ideas for all these results come from the similar characterizations of antimatroids and convex geometries, and due to the fact that both of them are special cases of shattering-extremal systems.