  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 $\mathbb{R}^n$.
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.
