  We prove several new results around Gromov's waist theorem. We give a simple
proof of Vaaler's theorem on sections of the unit cube using the
Borsuk--Ulam--Crofton technique. We consider waists of real and complex
projective spaces, flat tori, convex bodies in Euclidean space. We establish
waist-type results in terms of the Hausdorff measure.
