Ham Sandwich with Mayo: A Stronger Conclusion to the Classical Ham Sandwich Theorem

The conclusion of the classical ham sandwich theorem of Banach and Steinhaus may be strengthened: there always exists a common bisecting hyperplane that touches each of the sets, that is, intersects the closure of each set. Hence, if the knife is smeared with mayonnaise, a cut can always be made so that it will not only simultaneously bisect each of the ingredients, but it will also spread mayonnaise on each. A discrete analog of this theorem says that n finite nonempty sets in n-dimensional Euclidean space can always be simultaneously bisected by a single hyperplane that contains at least one point in each set. More generally, for n compactlysupported positive finite Borel measures in Euclidean n-space, there is always a hyperplane that bisects each of the measures and intersects the support of each measure.