Ham Sandwich Theorem
In n dimensions, one hyperplane can simultaneously bisect n measurable bodies.
Slice a ham sandwich — two slices of bread and a slab of ham — with one straight cut so each ingredient is exactly halved. The theorem guarantees this is always possible: given n measurable bodies in Rⁿ, a single hyperplane bisects all n simultaneously. The proof is a direct application of Borsuk–Ulam — each direction on Sⁿ⁻¹ determines a family of hyperplanes, and the topological obstruction forces a bisecting cut to exist.
Formula
One hyperplane can simultaneously bisect n finite Borel measures in Rⁿ — proved via the Borsuk–Ulam theorem.
Map each direction to the imbalance vector; Borsuk–Ulam forces a zero, which is the bisecting hyperplane.
The polynomial generalization uses algebraic hypersurfaces instead of hyperplanes, key to the Erdős distinct-distances solution.
Summary
Slice a ham sandwich — two slices of bread and a slab of ham — with one straight cut so each ingredient is exactly halved. The theorem guarantees this is always possible: given n measurable bodies in Rⁿ, a single hyperplane bisects all n simultaneously. The proof is a direct application of Borsuk–Ulam — each direction on Sⁿ⁻¹ determines a family of hyperplanes, and the topological obstruction forces a bisecting cut to exist.
