  We examine connections between combinatorial notions that arise in machine
learning and topological notions in cubical/simplicial geometry. These
connections enable to export results from geometry to machine learning. Our
first main result is based on a geometric construction by Tracy Hall (2004) of
a partial shelling of the cross-polytope which can not be extended. We use it
to derive a maximum class of VC dimension 3 that has no corners. This refutes
several previous works in machine learning from the past 11 years. In
particular, it implies that all previous constructions of optimal unlabeled
sample compression schemes for maximum classes are erroneous.
  On the positive side we present a new construction of an unlabeled sample
compression scheme for maximum classes. We leave as open whether our unlabeled
sample compression scheme extends to ample (a.k.a. lopsided or extremal)
classes, which represent a natural and far-reaching generalization of maximum
classes. Towards resolving this question, we provide a geometric
characterization in terms of unique sink orientations of the 1-skeletons of
associated cubical complexes.
