Ham Sandwich Theorem

The volumes of any $n$ $n$-D solids can always be simultaneously bisected by a $(n-1)$-D 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 theorem was proved for $n>3$ by Stone and Tukey (1942).

See also Pancake Theorem

References

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.

Stone, A. H. and Tukey, J. W. ``Generalized `Sandwich' Theorems.'' Duke Math. J. 9, 356-359, 1942.