Ham Sandwich Theorem

The volumes of any n n-dimensional solids can always be simultaneously bisected by a (n-1)-dimensional hyperplane. Proving the theorem for n=2 (where it is known as the pancake theorem) is simple and can be found in Courant and Robbins (1978).

The proof is more involved for n=3 (Hunter and Madachy 1975, p. 69), but an intuitive proof can be obtained by the following argument due to G. Beck (pers. comm., Feb. 18, 2005). Note that given any direction n^^, the volume of a solid can be bisected by a plane with normal n^^. To see this, start with a plane that has all of the solid on one side and move it parallel to itself until the solid is completely on its other side. There must have been an intermediate position where the plane bisected the solid.

Now take a sphere centered at the origin large enough to contain the three solids. Each point on the surface of the sphere indicates a direction. For any direction and each solid, find a plane that bisects the solid with that direction as normal. So each direction gives three planes parallel to each other. Define x and y to be the directed distances between one of the planes to each of the other two, and for each point on the sphere, associate a point in the xy-plane.

If P is opposite to Q on the sphere, the three planes for the direction P are the same as those for the direction Q. But the distances between the planes are directed, so the point (x(P),y(P)) is opposite (x(Q),y(Q) in the xy-plane.

As a point (direction) moves along a meridian from the north pole to the south pole and then back up the other side to the north pole again, the point (x,y) traces a closed curve in the xy-plane consisting of opposite points. It must therefore enclose the origin. Rotating the meridian a half turn makes the curve deform until it coincides with itself, but with points moving to their opposites. At some rotation of the meridian between "none" and "half a turn," the curve crosses the origin, x=0 and y=0, which means the three planes are one, a simultaneous bisector of the three solids.

The theorem was proved for n>3 by Stone and Tukey (1942).

See also

Cutting, Pancake Theorem

Portions of this entry contributed by George Beck

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Chinn, W. G. and Steenrod, N. E. First Concepts of Topology. Washington, DC: Math. Assoc. Amer., 1966.Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods. Oxford, England: Oxford University Press, 1978.Davis, P. J. and Hersh, R. The Mathematical Experience. Boston, MA: Houghton Mifflin, pp. 274-284, 1981.Hunter, J. A. H. and Madachy, J. S. Mathematical Diversions. New York: Dover, pp. 67-69, 1975.Steinhaus, H. "Sur la division des ensembles de l'espace par les plans et des ensembles plans par les cercles." Fundamenta Math. 33, 245-263, 1945.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 145, 1999.Stone, A. H. and Tukey, J. W. "Generalized 'Sandwich' Theorems." Duke Math. J. 9, 356-359, 1942.

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Ham Sandwich Theorem

Cite this as:

Beck, George and Weisstein, Eric W. "Ham Sandwich Theorem." From MathWorld--A Wolfram Web Resource.

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