Many years ago in 1985 I proved the following lemma that I used in proving a major theorem about

the stability of matter when I was at the Institute in Princeton as a post-doc. Can anyone reprove it? I have another lemma that I proved lets see if you guys can prove that too. Here is the first lemma:

Take a square it works for a cube in any dimension. Now play the following game. Lets take a square for example.

Bisect each side you get now 4 congruent mini squares. Call this Step 1. Each congruent mini square can be bisected further to get now 16 mini squares in step 2. Proceed so on. One gets at step n, 4^n squares.

Now remove N DISJOINT squares in any way, that is remove a step 10 square, a step 25 square etc. The remaining

figure is a big square with holes like Swiss cheese Prove you can cover the remaining figure; the Swiss cheese looking part by exactly C N squares. C is a universal constant that only depends on the dimension. I forget the value I found for C for dimension 2.

By cover I mean nothing must penetrate the holes where squares were taken out.

The paper I wrote appears in Communications in Partial Differential Equations towards the end of the paper.

Another cute geometric lemma I proved in 1991 appears in a paper in Journal of Differential Geometry.

Only recently has more progress been made on it. Here is the lemma, actually can anybody improve the lemma?

Here is the lemma. Take ANY set E in n-dimensions, n>1. Assume it is covered by finitely many balls of various radii, radii is all arbitrary. Covered means every point in E is in some ball. Now prove one can find a sub family of the balls that have the following property:

(1) For any POSITIVE number delta<1/2, if I magnify the radii of each ball of the sub-family by 1+\delta, i.e.

new radius is (1+delta)r, then E is still covered by the balls of the sub-family with the radius of each magnified by 1+delta

(2) FURTHERMORE, a point does not belong to more than C (delta)^{-n} balls of the sub-family, where C is a UNIVERSAL constant depending only on the dimesnion n. That is in other words the sub-family without the radius magnification is more or less disjoint, there is overlap between balls but the overlap is controlled, i.e no more than C{delta}^{-n} balls of the sub-family overlap.

This lemma is used by me to obtain a powerful result a conjecture of ST Yau. Point is can you improve statement

(2) to (delta)(- {\square root n}} for example, any improvement?

By the way there is a new paper by Dan Mangoubi on arxiv.org who is claiming to have improved my technique.

I have not checked his computations. There is a huge amount of work that came out of the innocuous lemma 2.

Here is one which eventually appeared in the American Journal of Mathematics one of the leading journals of the world in 2004.

http://arxiv.org/abs/math/0402412You will be able to do better than what the authors above do if you improve the second lemma since you will improve my result and its ensuing major consequences.