Lifting methods in mass partition problems

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