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

TL;DR

This paper strengthens the classical ham sandwich theorem by proving the existence of a bisecting hyperplane that also intersects the closure of each set, with applications to measures and discrete sets.

Contribution

It introduces a stronger version of the ham sandwich theorem, showing the hyperplane can touch each set or measure, not just bisect them, extending classical results.

Findings

Existence of a hyperplane intersecting the closure of each set

Extension to measures with supports intersected by the hyperplane

Application to planetary and lunar mass bisecting planes

Abstract

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 compactly-supported 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. For example, at any given instant of time, there is one planet, one moon and one asteroid in our solar system and a single plane touching all three that exactly bisects the total planetary mass, the total lunar mass, and the total asteroidal mass of the solar system.