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Lutwak's volume inequalities for polar projection bodies of all orders aregeneralized to polarizations of Minkowski valuations generated by even, zonalmeasures on the Euclidean unit sphere. This is based on analogues of mixedprojection bodies for such Minkowski valuations and a generalization of thenotion of centroid bodies. A new integral representation is used to single outLutwak's inequalities as the strongest among these families of inequalities,which in turn are related to a conjecture on affine quermassintegrals. In thedual setting, a generalization of volume inequalities for intersection bodiesof all orders by Leng and Lu is proved. These results are related to Grinberg'sinequalities for dual affine quermassintegrals.

Lutwak-Petty projection inequalities for Minkowski valuations and their duals