All upper semicontinuous and SL(n) invariant valuations on convex bodies containing the origin in their interiors are completely classified. Each such valuation is shown to be a linear combination of the Euler characteristic, the volume, the volume of the polar body, and the recently discovered Orlicz surface areas.Mathematics subject classification: 52A20, 52B45

All SL(n)-contravariant L p -Minkowski valuations on polytopes are completely classified. The prototypes of such valuations turn out to be the asymmetric L p -projection body operators.

A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn–Minkowski type inequalities for rigid motion intertwining Minkowski valuations.

We consider valuations defined on polytopes containing the origin which have measures on the sphere as values. We show that the classical surface area measure is essentially the only such valuation which is SL(n) contravariant of degree one. Moreover, for all real p, an L p version of the above result is established for GL(n) contravariant valuations of degree p. This provides a characterization of the L p surface area measures from the L p Brunn-Minkowski theory.