Let

*X,Y,Z* be the three

bounded subsets of

**R**^{3}
in the statement of the

ham sandwich theorem. We can assume that they
lie inside a

sphere *S* with

radius 1/2. Now choose a

point *p*
on the sphere. There is a

line segment joining it (via the sphere's centre)
to its

antipodal point. For each s in the interval [0,1] consider the

plane
*P*_{s} which is perpendicular to this line segment and meets
the line segment at a point which is a distance

*s* from

*p*.
Obviously

*P*_{s} breaks up

*X* into two pieces,

*X*_{near} and

*X*_{far}, with

*X*_{near} being the closest to

*p*.

Define a function *v:[0,1]->***R** by
*v(s)*=volume(*X*_{near})-volume(*X*
_{far})
Think of the plane *P*_{s} moving from *p* to its
antipodal point as *s* varies. As it does so, the volume of
the near subset gets bigger and the volume of the far subset gets smaller.
So we see that *v* is monotonically increasing. Also
*v(0)* is the volume of *X* and *v(1)* is the negative
of this value. So the Intermediate Value Theorem tells us immediately
that *v* vanishes at some point of *[0,1]*. Because *v*
increases this means that it either *vanishes* at exactly one point
or at a closed interval. Define a function *x:S->***R**
by assigning to *p* the vanishing point or the midpoint of the vanishing
interval. *x* is continuous and has *x(-s)=1-x(s)*
Note that *P*_{x(p)} divides *X* exactly in half. Likewise
we define *y,z*, for *Y,Z* in exactly the same way.

Define *f:S->***R**^{2} by
*f(p)=(x(p)-y(p),x(p)-z(p))*.
I claim that *f* vanishes at some point *p*. If it does then the theorem
is proved because this means that *x(p)=y(p)=z(p)*. Thus the plane
*P*_{x(p)} will divide *X,Y* and *Z* exactly
in half.

Suppose not then define *g:S->*S^{1} by *g(p)=f(p)/||f(p)||*.
This is continuous and has *g(p)=g(-p)*, for each *p*, contradicting
the Borsuk-Ulam Theorem.