Our method for getting lower bounds for 
is based
on cutting M up into infinitely skinny tubes.
In other applications it will be most convenient to consider
tubes modeled on 
that begin inside the manifold
and run out to infinity in one direction.
Here, we will consider only tubes modeled on 
that run out
to infinity in both directions.
There are two excuses for this.
The first is that these doubly-infinite tubes are
best for the specific application we have in mind.
The second is that you can always get a singly-infinite
tube by folding a doubly-infinite tube in half.
Definition. A cutting of M into tubes consists of a measure space T (to index the tubes) and measurable maps
(to show how they run, and tell their cross-section) such that
on 
over onto the Riemannian volume measure on M,
that is,
for any integrable function u on M,
we have
(This makes precise the notion that 
tells the cross-section of
the tube.)
, the curve 
is
piecewise 
, parametrized by arc length, and proper.
(The tubes may zig-zag a little, but must run out to the boundary.) we have
whenever 
.
(The tubes are neither too fat nor too thin.)