Affine quermassintegrals and even Minkowski valuations

It is shown that each continuous even Minkowski valuation on convex bodies of degree $1\le i\le n-1$ intertwining rigid motions is obtained from convolution of the ith projection function with a unique spherical Crofton distribution. In case of a non-negative distribution, the polar volume of the associated Minkowski valuation gives rise to an isoperimetric inequality which strengthens the classical relation between the ith quermassintegral and the volume. This large family of inequalities unifies earlier results obtained for $i=1$ and $n-1$. In these cases, isoperimetric inequalities for affine quermassintegrals, specifically the Blaschke–Santaló inequality for $i=1$ and the Petty projection inequality for $i=n-1$, were proven to be the strongest inequalities. An analogous result for the intermediate degrees is established here. Finally, a new sufficient condition for the existence of maximizers for the polar volume of Minkowski valuations intertwining rigid motions reveals unexpected examples of volume inequalities having asymmetric extremizers.