  We study the algebraic boundary of a convex semi-algebraic set via duality in
convex and algebraic geometry. We generalize the correspondence of facets of a
polytope to the vertices of the dual polytope to general semi-algebraic convex
bodies. In the general setup, exceptional families of extreme points might
exist and we characterize them semi-algebraically. We also give an algorithm to
compute a complete list of exceptional families, given the algebraic boundary
of the dual convex set.
