Ham Sandwich is Equivalent to Borsuk-Ulam

The Borsuk-Ulam theorem is a fundamental result in algebraic topology, with applications to various areas of Mathematics. A classical application of the Borsuk-Ulam theorem is the Ham Sandwich theorem: The volumes of any n compact sets in R can always be simultaneously bisected by an (n− 1)-dimensional hyperplane. In this paper, we demonstrate the equivalence between the Borsuk-Ulam theorem and the Ham Sandwich theorem. The main technical result we show towards establishing the equivalence is the following: For every odd polynomial restricted to the hypersphere f : S → R, there exists a compact set A ⊆ Rn+1, such that for every x ∈ S we have f(x) = vol(A∩H+)−vol(A∩H−), where H is the oriented hyperplane containing the origin with ~x as the normal. A noteworthy aspect of the proof of the above result is the use of hyperspherical harmonics. Finally, using the above result we prove that there exist constants n0, ε0 > 0 such that for every n ≥ n0 and ε ≤ ε0/ √ 48n, any query algorithm to find an ε-bisecting (n − 1)-dimensional hyperplane of n compact sets in [−n4.51, n4.51]n, even with success probability 2−Ω(n), requires 2Ω(n) queries. 1998 ACM Subject Classification F.2.2 Computations on discrete structures, Geometrical problems and computations