The volumes of any -dimensional solids can always be simultaneously
bisected by a -dimensional
hyperplane. Proving the theorem for (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 (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 , the volume of a solid can be bisected by a plane with normal
.
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 and 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
-plane.

If
is opposite to
on the sphere, the three planes for the direction are the same as those for the direction . But the distances between the planes are directed, so the
point
is opposite
in the -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 traces a closed curve in the -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,
and ,
which means the three planes are one, a simultaneous bisector of the three solids.

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

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.