Norm estimates for k-plane transforms and geometric inequalities

The article is devoted to remarkable interrelation between the norm estimates for k-plane transforms in weighted and unweighted L-P spaces and geometric integral inequalities for cross-sections of measurable sets in R-n. 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. (C) 2019 Elsevier Inc. All rights reserved.