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).
