Equitable 3-cuttings in two and three dimensions

for the plane that states that, for finite sets of red and blue points
in the plane, there exists a line dividing both red and blue points
into sets of equal size. We consider equitable 3-cuttings of 2 arbitrary
mass distributions in the plane (partitions of the plane into 3 sectors
with a common apex such that each sector contains 1/3 of the 2 mass
distributions). We prove the existence of a continuum of equitable
3-cuttings that satisfy some closure property. This permits us to
generalize earlier results on both convex and non-convex equitable
3-cuttings with additional constraints. It is also used to prove
the existence of an equitable 3-cutting of 3 masses in 3 dimensions.

We present algorithms to compute equitable 3-cuttings in two and three
dimensions for discrete versions of problems