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We provide general inequalities that compare the surface area
of a convex body
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
. We examine separately the dependence of the constants on the dimension in the case where
is in some of the classical positions or
is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.
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