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.

Log in or register to write something here or to contact authors.