Norm Estimates for k-Plane Transforms and Geometric Inequalities

الأصل · EN

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. We also consider 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.