My proof of the Geometry-Real Analysis theorem

In summary, the conversation discusses a convex shape inside a unit square and the supremum of subsets that can be obtained from scaled and translated copies of the shape. The square is partitioned into smaller squares of three types: exterior, interior, and boundary. The conversation then introduces an equation and uses it to prove that finitely many disjoint discs can be inscribed in a unit square with a total area approaching 1. Finally, there is a question about the correctness of the proof and a request for justification for a specific step.
  • #1
Mike400
59
6
Homework Statement
My proof of the theorem : finitely many disjoint discs can be inscribed in a unit square with total area approaching 1
Relevant Equations
$$\dfrac{i}{n^2} + \dfrac{b}{n^2} < A + \dfrac{b}{n^2} \tag1$$
$$\text{Area of finite disc packing} = e \dfrac{a- \epsilon}{n^2} + A $$
$$\left[ 1 - A - \dfrac{b}{n^2} \right] (a- \epsilon) + A<a$$
Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##.

Partition the square into ##n×n## smaller squares (see picture).
There are three types of such small squares: ##e## exterior squares (white in the picture), ##i## interior squares (light red in the image) and ##b## boudary squares (blue/purple). Of course ##e+b+i=n^2##

1671786017908.png


$$\dfrac{i}{n^2} < A$$
$$\implies\dfrac{i}{n^2} + \dfrac{b}{n^2} < A + \dfrac{b}{n^2} \tag1$$

Picking a finite packing that covers ##\ge a-\epsilon##, for some ##\epsilon##, we can put a scaled-down copy of this packing into each of the ##e## "white" squares and, together with the original shape ##S##, obtain a finite packing of the unit square that covers ##e \dfrac{a- \epsilon}{n^2} + A##. By using this fact and equation ##(1)##

$$a>\text{Area of finite disc packing}
= e \dfrac{a- \epsilon}{n^2} + A$$
$$= \dfrac{e}{n^2} (a- \epsilon) + A
= \left[ 1- \left( \dfrac{i}{n^2} + \dfrac{b}{n^2} \right) \right] (a- \epsilon) + A $$
$$\geq \left[ 1- \left( A + \dfrac{b}{n^2} \right) \right] (a- \epsilon) + A$$
$$=\left[ 1 - A - \dfrac{b}{n^2} \right] (a- \epsilon) + A $$

$$\implies \left[ 1 - A - \dfrac{b}{n^2} \right] (a- \epsilon) + A<a$$

As ##n\to\infty## and ##\epsilon\to 0## the LHS converges to ##a+(1-a)A##. According to a limit theorem, this limit must be ##\le a##. Thus we conclude ##a=1##.

Thus we have shown : finitely many disjoint discs can be inscribed in a unit square with total area approaching 1.

My question : Is my proof of the theorem correct?
 
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  • #2
I don't understand the last step, what happened to e for example?

Edit: I reread it and get it better now. My brain failed to string together the lines correctly.

I feel like you probably need to justify why ##b/n^2\to 0##. Other than that it seems fine.

@Mike400 pinging for visibility.
 
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FAQ: My proof of the Geometry-Real Analysis theorem

1. What is the Geometry-Real Analysis theorem?

The Geometry-Real Analysis theorem is a mathematical theorem that establishes a relationship between geometric figures and real numbers. It states that any geometric figure can be represented by a set of real numbers and vice versa.

2. What is the significance of this theorem?

This theorem has significant implications in both geometry and real analysis. It allows for a deeper understanding of the connection between the two fields and provides a powerful tool for solving complex mathematical problems.

3. How was this theorem proven?

The proof of the Geometry-Real Analysis theorem involves using a combination of geometric and algebraic techniques. It requires a deep understanding of both fields and involves rigorous logical reasoning.

4. Can this theorem be applied in real-life situations?

Yes, the Geometry-Real Analysis theorem has many practical applications. For example, it can be used in engineering and architecture to design and analyze structures, as well as in computer graphics to create realistic 3D models.

5. Are there any limitations to this theorem?

Like any mathematical theorem, the Geometry-Real Analysis theorem has its limitations. It may not be applicable to all types of geometric figures or real numbers, and its proof may not be easily generalized to other mathematical concepts.

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