The general case of the Ham-Sandwich Theorem says that if we have

The general proof is suggested by the argument in the two-dimensional case.
There, for each possible direction **s** for the cut, we clearly have,
for each region, a line in direction **s** bisecting that region.
But the two lines for the two regions are offset by some distance **d(s)**.
We'd like to find a direction with **d(s) = 0**.

Note that, if we have rotate our direction by 180 degrees,
we get back to the same pair of bisecting lines,
but they now have the opposite orientation. Adopting the convention
that the distance **d** between the lines is a signed quantity
depending on the orientation, we see that **d(s+180) = -d(s)**.
Thus it is clear that **d** is neither always positive nor always
negative. Since **d** is a continuous function,
by the intermediate-value theorem it must achieve a value of **0**
for some direction.

Thinking of the circle of directions as the unit circle
in the plane, we might write **-s** instead of **s+180** for the
opposite direction.
Thus we have a function **d** from the circle of
directions to the real numbers, with the property that antipodal points map
to negatives of each other: **d(-s) = -d(s)**.
Such a function must be zero somewhere.

In three dimensions, we can think of a direction **s**
as a point on the sphere **S ^{2}**;
given

Again, if we move from direction **s** to the opposite direction
**-s** (the antipodal point on the sphere), the bisecting planes
will be the same, but with opposite orientation. Thus each
of the distances again gets negated. That is, we have a function
from **S ^{2}** to

The Ham-Sandwich theorem claims that this function must map some point on the sphere to the origin. This is a two-dimensional analog of the intermediate-value theorem; it is the two-dimensional case of the Borsuk-Ulam theorem.

The Borsuk-Ulam theorem in general dimensions can be stated in a number of
ways but always deals with a map **d** from sphere to sphere or
from sphere to euclidean space which is *odd*,
meaning that **d(-s)=-d(s)**.
Another way to describe this property is to say that **d** is
equivariant with respect to the antipodal map (negation).

A nice discussion of the Borsuk-Ulam theorem
can be found in Section 2.6 of Guillemin and Pollack's
book *Differential Topology*. Here we will explain the
different formulations, and their connections with each other and
with the Ham-Sandwich theorem.

One formulation of Borsuk-Ulam says that an odd map
from **S ^{n}** to

This means (by definition) exactly that, if we replace **d** by the map
**d/|d|** from **S ^{n}** to itself (which will also be odd),
it will have odd degree. (The mod-2

An immediate corollary is that there is no odd
map from **S ^{n}** to

An equivalent corollary says that an odd map from **S ^{n}** to

Another statement of the corollary says that given any function
**f** from **S ^{n}** to

This last corollary has interesting consequences.
For instance, at any time there is some pair of antipodal points
on earth where the temperature and pressure are exactly the same.
It is also a nice way to see that the
**n**-sphere is not homeomorphic to any subset of euclidean **n**-space.
For instance there is no way to make a (one-to-one) map of the earth
on a planar piece of paper without ripping the image somewhere.

As was suggested by the discussion in two and three dimensions,
it is the second corollary
(that any odd map from **S ^{n-1}** to

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