Faces of convex sets generalise the notion of faces of
polytopes (such as vertices, edges and facets). Many structural
properties of convex sets can be expressed via faces and their
arrangements. In particular, some regularity conditions that guarantee
good performance of optimisation methods can be expressed in terms of
facial structure.
I will mention some classical and recent results related to the facial
structure of convex sets, and will illustrate the impact of facial
structure on the performance of numerical methods using the recent
counter-example to De Pierro's conjecture about the convergence of
under-related cyclic projections.