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. The proof is more involved for n = 3, but an intuitive proof can be obtained by the following argument due to G. Beck. 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.

cutting | pancake theorem

sandwich