**EggMath: The White/Yolk Theorem**

**The Borsuk-Ulam Theorem**

The general case of the Ham-Sandwich Theorem
says that if we have **n** regions in
**n**-dimensional space, then there is some
hyperplane which cuts each exactly in half,
measured by volume.

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.

**Three dimensions**

In three dimensions, we can think of a
direction **s** as a point on the sphere
**S**^{2}; given **s** we cut each
of our regions in half with a plane normal to the
ray from the origin to **s**. With three
regions, there are now three bisecting planes, and
thus two distances between them. (It doesn't
really matter which pairs we pick to measure
distance between.) Thus we have a function
**d** from the two-sphere to the plane (pairs
of real numbers).

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 **R**^{2} with the property that
antipodal points map to negatives of each other:
**d(-s) = -d(s)**.

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.

**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
**R**^{n+1} whose image does not
contain the origin must in fact have nonzero mod-2
winding number around the origin.

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
*degree* of a map between compact manifolds
is the number of preimages of a generic point.)
Thus an equivalent statement of the theorem is
that any odd map from **S**^{n} to
itself has odd degree, and thus in particular will
be onto.

An immediate corollary is that there is no odd
map from **S**^{n} to
**S**^{n-1}. For any such map could be
composed with the inclusion of
**S**^{n-1} into **S**^{n}
to give an odd map whose image was contained in
the equator.

An equivalent corollary says that an odd map
from **S**^{n} to **R**^{n}
must have the origin as a value somewhere. For if
not, we could replace **d** by **d/|d|** and
get an odd map to **S**^{n-1}.

Another statement of the corollary says that
given any function **f** from
**S**^{n} to **R**^{n},
there is a pair of antipodal points where the
function values are equal. To prove this, let
**d(s) = f(s) - f(-s)**.

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
**R**^{n-1} must have the origin as a
value somewhere) which proves the Ham-Sandwich
Theorem. For if we have **n** regions in
**R**^{n}, then given a direction
**s** in **S**^{n-1}, we can find
planes in that direction which slice the regions
in half. If **d** measures the **n-1**
signed distances between these planes, then
**d** is an odd map from **S**^{n-1}
to **R**^{n-1}.

The usual proof of
the theorem is by induction.

**Caveats**

We have not been careful about the class of
maps we consider in this discussion. Certainly
all the theorems are true for smooth (infinitely
differentiable) maps, as in Guillemin/Pollack. We
also need to be careful, when constructing the map
**d** used to prove the Ham-Sandwich theorem,
to even ensure that it is continuous. If some
region consists of two separate pieces of equal
volume, then for some directions, there will be a
choice of several hyperplanes in that direction
which bisect the region. If we made random
choices, our function **d** would be
discontinuous. However, if we choose the planes
carefully, we do get a smooth function.