and 
In estimating 
for our tubes,
we will want to know what effect altering and
by some bounded factor will have on 
.
There are two reasons for this.
The first is that when all we are after is a very conservative
lower bound for ,
we may want to hack the space into tubes
in such a way that we have only very gross information about
the growth of the cross-section of the tubes.
The second reason is that to get a lower bound for ,
it may be convenient to assume that and are pretty smooth.
Lemma.
For any K
( 
),
Lemma.
Let
be measurable,
with 
and 
neither too big nor too small.
If
and
then
Lemma.
Let
be measurable,
with and neither too big nor too small.
Suppose that for constants 
( 
),
we have
and
Then
