Common hyperplane medians for random variables
American Mathematical Monthly
On generalizations of Radon's theorem and the Ham sandwich theorem
European Journal of Combinatorics - Special issue dedicated to Bernt Lindstro¨m
2-Dimension Ham Sandwich Theorem for Partitioning into Three Convex Pieces
JCDCG '98 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
Slicing Convex Sets and Measures by a Hyperplane
Discrete & Computational Geometry
Uneven Splitting of Ham Sandwiches
Discrete & Computational Geometry
Discrete & Computational Geometry
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The conclusion of the classical ham sandwich theorem for bounded Borel sets may be strengthened, without additional hypotheses-there always exists a common bisecting hyperplane that touches each of the sets, that is, that intersects the closure of each set. In the discrete setting, where the sets are finite (and the measures are counting measures), there always exists a bisecting hyperplane that contains at least one point in each of the sets. Both these results follow from the main theorem of this note, which says that for n compactly supported positive finite Borel measures in R^n, there is always an (n-1)-dimensional hyperplane that bisects each of the measures and intersects the support of each measure. Thus, 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. In contrast to the bisection conclusion of the classical ham sandwich theorem, this bisection-and-intersection conclusion does not carry over to unbounded sets of finite measure.