  We provide general inequalities that compare the surface area S(K) of a
convex body K in ${\mathbb R}^n$ to the minimal, average or maximal surface
area of its hyperplane or lower dimensional projections. We discuss the same
questions for all the quermassintegrals of K. We examine separately the
dependence of the constants on the dimension in the case where K is in some of
the classical positions or K is a projection body. Our results are in the
spirit of the hyperplane problem, with sections replaced by projections and
volume by surface area.
