  This work studies embedding of arbitrary VC classes in well-behaved VC
classes, focusing particularly on extremal classes. Our main result expresses
an impossibility: such embeddings necessarily require a significant increase in
dimension. In particular, we prove that for every $d$ there is a class with VC
dimension $d$ that cannot be embedded in any extremal class of VC dimension
smaller than exponential in $d$.
  In addition to its independent interest, this result has an important
implication in learning theory, as it reveals a fundamental limitation of one
of the most extensively studied approaches to tackling the long-standing sample
compression conjecture. Concretely, the approach proposed by Floyd and Warmuth
entails embedding any given VC class into an extremal class of a comparable
dimension, and then applying an optimal sample compression scheme for extremal
classes. However, our results imply that this strategy would in some cases
result in a sample compression scheme at least exponentially larger than what
is predicted by the sample compression conjecture.
  The above implications follow from a general result we prove: any extremal
class with VC dimension $d$ has dual VC dimension at most $2d+1$. This bound is
exponentially smaller than the classical bound $2^{d+1}-1$ of Assouad, which
applies to general concept classes (and is known to be unimprovable for some
classes). We in fact prove a stronger result, establishing that $2d+1$ upper
bounds the dual Radon number of extremal classes. This theorem represents an
abstraction of the classical Radon theorem for convex sets, extending its
applicability to a wider combinatorial framework, without relying on the
specifics of Euclidean convexity. The proof utilizes the topological method and
is primarily based on variants of the Topological Radon Theorem.
