| Let X,Y,Z be the three bounded subsets of R3
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
Ps which is perpendicular to this line segment and meets
the line segment at a point which is a distance s from p.
Obviously Ps breaks up X into two pieces,
Xnear and Xfar, with
Xnear being the closest to p.
Define a function v:[0,1]->R by
v(s)=volume(Xnear)-volume(X
far)
Think of the plane Ps 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 Px(p) divides X exactly in half. Likewise
we define y,z, for Y,Z in exactly the same way.
Define f:S->R2 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
Px(p) will divide X,Y and Z exactly
in half.
Suppose not then define g:S->S1 by g(p)=f(p)/||f(p)||.
This is continuous and has g(p)=g(-p), for each p, contradicting
the Borsuk-Ulam Theorem. | 
