Equipartition of a continuous mass distribution

Here are three samples of results. (1) Let m be a finite (absolutely) continuous mass distribution in ℝ², and let ℓ = {ℓ₁, ..., ℓ₅ ⊂ ℝ²} be a quintuple of rays with common origin such that any two adjacent angles between them make a sum of at most π. Then an affine image of ℓ subdivides m into five parts with any prescribed ratios. (2) For each finite continuous mass distribution m in ℝⁿ, there exist n mutually orthogonal hyperplanes any two of which quarter m. (3) Let m and m′ be two finite continuous mass distributions in ℝRⁿ with common center of symmetry O. Then there exist n hyperplanes through O any two of which quarter both m and m′. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 92–106.