Abstract
The article is devoted to remarkable interrelation between the norm estimates for k-plane transforms in weighted and unweighted Lp spaces and geometric integral inequalities for cross-sections of measurable sets in Rn. A similar interrelation is studied for more general j-plane to k-plane transforms on affine Grassmannians and their compact modifications. The article contains a series of new integral-geometric inequalities with sharp constants, explicit equalities, conjectures, and open problems.
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