  Algebraic boundaries of convex semi-algebraic sets are closely related to
polynomial optimization problems. Building upon Rainer Sinn's work, we refine
the stratification of iterated singular loci to a Whitney (a) stratification,
which gives a list of candidates of varieties whose dual is an irreducible
component of the algebraic boundary of the dual convex body. We also present an
algorithm based on Teissier's criterion to compute Whitney (a) stratifications,
which employs conormal spaces and prime decomposition.
