Lifting Methods in Mass Partition Problems

Abstract Many results about mass partitions are proved by lifting $\mathds {R}^d$ to a higher-dimensional space and dividing the into pieces. We extend such methods use arguments polyhedral surfaces. Among other results, we prove existence of equipartitions $d+1$ measures in parallel hyperplanes $d+2$ concentric spheres. For whose supports sufficiently well separated, where one can cut fixed (possibly different) fraction each measure either hyperplanes, spheres, convex surfaces few facets, or polytopes with vertices.