On the Blaschke-Petkantschin Formula and Drury's Identity

The Blaschke-Petkantschin formula is a variant of the polar decomposition of the $k$-fold Lebesgue measure on $\mathbb {R}^n$ in terms of the corresponding measures on $k$-dimensional linear subspaces of $\mathbb {R}^n$. We suggest a new elementary proof of this formula and discuss its connection with the celebrated Drury's identity that plays a key role in the study of mapping properties of the Radon-John $k$-plane transforms. We give a new derivation of this identity and provide it with precise information about constant factors and the class of admissible functions.