  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.
