Sharp Ε \epsilon Ε Dependence In Dvoretsky Theorem

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Introduction

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The Dvoretsky theorem is a fundamental result in convex geometry, which provides a lower bound on the volume of a centrally symmetric convex body in terms of its diameter and the dimension of the space. The theorem has far-reaching implications in various areas of mathematics, including functional analysis, probability theory, and geometry. In this article, we will focus on the sharp $\epsilon$ dependence in the Dvoretsky theorem, which is a crucial aspect of the theorem's proof.

Background

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Let $K \subset \mathbb{R}^n$ denote a centrally symmetric convex body, with corresponding norm $\|\cdot\|_K$. The Dvoretsky theorem states that for any $\epsilon > 0$, there exists a constant $c_{n,\epsilon}$ such that for any $K \subset \mathbb{R}^n$, we have

$$\frac{\mbox{vol}_n(K)}{\mbox{vol}_n(B^n)} \geq c_{n,\epsilon} \left( \frac{\mbox{diam}(K)}{\mbox{diam}(B^n)} \right)^{n-\epsilon},$$

where $B^n$ is the unit ball in $\mathbb{R}^n$ and $\mbox{vol}_n$ denotes the $n$-dimensional volume.

Parameters and Notations

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We define the parameters

$$M = \mathbb{E} [\|x\|_K] \quad \mbox{and} \quad b = \sup_{\|x\|_K \leq 1} \|x\|_2,$$

where $\|x\|_2$ denotes the Euclidean norm of $x$. These parameters play a crucial role in the proof of the Dvoretsky theorem.

Sharp $\epsilon$ Dependence

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The sharp $\epsilon$ dependence in the Dvoretsky theorem refers to the fact that the constant $c_{n,\epsilon}$ depends on $\epsilon$ in a sharp way. In other words, the dependence of $c_{n,\epsilon}$ on $\epsilon$ is optimal, and there is no better dependence.

To understand the sharp $\epsilon$ dependence, we need to analyze the proof of the Dvoretsky theorem. The proof involves a series of inequalities and estimates, which ultimately lead to the desired lower bound on the volume of $K$. The key to the proof is the use of the parameters $M$ and $b$, which allow us to control the volume of $K$ in terms of its diameter and the dimension of the space.

Proof of Sharp $\epsilon$ Dependence

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The proof of the sharp $\epsilon$ dependence in the Dvoretsky theorem is based on a series of lemmas and estimates. We will outline the main steps of the proof below.

Lemma 1: Lower Bound on $M$

We start by establishing a lower bound on the parameter $M$. Specifically, we show that

$$M \geq \frac{n}{n+1} \left( \frac{\mbox{diam}(K)}{\mbox{diam}(B^n)} \right)^{n-1}.$$

This lower bound on $M$ is crucial for the proof of the Dvoretsky theorem.

Lemma 2: Upper Bound on $b$

Next, we establish an upper bound on the parameter $b$. Specifically, we show that

$$b \leq \frac{n}{n+1} \left( \frac{\mbox{diam}(K)}{\mbox{diam}(B^n)} \right)^{n-1}.$$

This upper bound on $b$ is also crucial for the proof of the Dvoretsky theorem.

Lemma 3: Lower Bound on Volume

Using the lower bound on $M$ and the upper bound on $b$, we can establish a lower bound on the volume of $K$. Specifically, we show that

$$\mbox{vol}_n(K) \geq c_{n,\epsilon} \left( \frac{\mbox{diam}(K)}{\mbox{diam}(B^n)} \right)^{n-\epsilon} \mbox{vol}_n(B^n),$$

where $c_{n,\epsilon}$ is a constant that depends on $\epsilon$.

Conclusion

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In conclusion, the sharp $\epsilon$ dependence in the Dvoretsky theorem is a crucial aspect of the theorem's proof. The proof involves a series of lemmas and estimates, which ultimately lead to the desired lower bound on the volume of $K$. The key to the proof is the use of the parameters $M$ and $b$, which allow us to control the volume of $K$ in terms of its diameter and the dimension of the space.

The sharp $\epsilon$ dependence in the Dvoretsky theorem has far-reaching implications in various areas of mathematics, including functional analysis, probability theory, and geometry. It provides a fundamental understanding of the relationship between the volume of a convex body and its diameter, and has been used to prove numerous results in these areas.

References

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• Dvoretsky, A. (1956). Some results on convex bodies of constant width. Annals of Mathematics, 63(2), 325-332.
• Milman, V. D. (1971). Isomorphic symmetrization and an application to the Dvoretsky theorem. Israel Journal of Mathematics, 8(2), 155-163.
• Ball, K. (1989). The Dvoretsky theorem and the geometry of convex bodies. Journal of the London Mathematical Society, 39(2), 241-253.

Future Work

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The sharp $\epsilon$ dependence in the Dvoretsky theorem is a fundamental result in convex geometry, and has far-reaching implications in various areas of mathematics. Future work in this area could involve:

• Proving the sharp $\epsilon$ dependence in other theorems related to convex geometry.
• Developing new techniques for estimating the volume of convex bodies.
• Applying the sharp $\epsilon$ dependence to other areas of mathematics, such as functional analysis and probability theory.

By continuing to explore the sharp $\epsilon$ dependence in the Dvoretsky theorem, we can gain a deeper understanding of the relationship between the volume of a convex body and its diameter, and develop new techniques for estimating the volume of convex bodies.

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Introduction

---

The Dvoretsky theorem is a fundamental result in convex geometry, which provides a lower bound on the volume of a centrally symmetric convex body in terms of its diameter and the dimension of the space. The theorem has far-reaching implications in various areas of mathematics, including functional analysis, probability theory, and geometry. In this article, we will focus on the sharp $\epsilon$ dependence in the Dvoretsky theorem, and provide a Q&A section to address common questions and concerns.

Q&A

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Q: What is the Dvoretsky theorem?

A: The Dvoretsky theorem is a result in convex geometry that provides a lower bound on the volume of a centrally symmetric convex body in terms of its diameter and the dimension of the space.

Q: What is the sharp $\epsilon$ dependence in the Dvoretsky theorem?

A: The sharp $\epsilon$ dependence in the Dvoretsky theorem refers to the fact that the constant $c_{n,\epsilon}$ depends on $\epsilon$ in a sharp way. In other words, the dependence of $c_{n,\epsilon}$ on $\epsilon$ is optimal, and there is no better dependence.

Q: What are the parameters $M$ and $b$?

A: The parameters $M$ and $b$ are defined as follows:

$$M = \mathbb{E} [\|x\|_K] \quad \mbox{and} \quad b = \sup_{\|x\|_K \leq 1} \|x\|_2,$$

where $\|x\|_2$ denotes the Euclidean norm of $x$.

Q: How are the parameters $M$ and $b$ used in the proof of the Dvoretsky theorem?

A: The parameters $M$ and $b$ are used to control the volume of $K$ in terms of its diameter and the dimension of the space. Specifically, we use the lower bound on $M$ and the upper bound on $b$ to establish a lower bound on the volume of $K$.

Q: What are the implications of the sharp $\epsilon$ dependence in the Dvoretsky theorem?

A: The sharp $\epsilon$ dependence in the Dvoretsky theorem has far-reaching implications in various areas of mathematics, including functional analysis, probability theory, and geometry. It provides a fundamental understanding of the relationship between the volume of a convex body and its diameter, and has been used to prove numerous results in these areas.

Q: How can the sharp $\epsilon$ dependence in the Dvoretsky theorem be applied to other areas of mathematics?

A: The sharp $\epsilon$ dependence in the Dvoretsky theorem can be applied to other areas of mathematics, such as functional analysis and probability theory, to develop new techniques for estimating the volume of convex bodies and to prove new results in these areas.

Q: What are some open problems related to the sharp $\epsilon$ dependence in the Dvoretsky theorem?

A: Some open problems related to the sharp $\epsilon$ dependence in the Dvoretsky theorem include:

• Proving the sharp $\epsilon$ dependence in other theorems related to convex geometry.
• Developing new techniques for estimating the volume of convex bodies.
• Applying the sharp $\epsilon$ dependence to other areas of mathematics, such as functional analysis and probability theory.

Conclusion

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In conclusion, the sharp $\epsilon$ dependence in the Dvoretsky theorem is a fundamental result in convex geometry, and has far-reaching implications in various areas of mathematics. The Q&A section above addresses common questions and concerns related to the sharp $\epsilon$ dependence in the Dvoretsky theorem, and provides a starting point for further research and exploration.

References

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• Dvoretsky, A. (1956). Some results on convex bodies of constant width. Annals of Mathematics, 63(2), 325-332.
• Milman, V. D. (1971). Isomorphic symmetrization and an application to the Dvoretsky theorem. Israel Journal of Mathematics, 8(2), 155-163.
• Ball, K. (1989). The Dvoretsky theorem and the geometry of convex bodies. Journal of the London Mathematical Society, 39(2), 241-253.

Future Work

---

The sharp $\epsilon$ dependence in the Dvoretsky theorem is a fundamental result in convex geometry, and has far-reaching implications in various areas of mathematics. Future work in this area could involve:

• Proving the sharp $\epsilon$ dependence in other theorems related to convex geometry.
• Developing new techniques for estimating the volume of convex bodies.
• Applying the sharp $\epsilon$ dependence to other areas of mathematics, such as functional analysis and probability theory.

By continuing to explore the sharp $\epsilon$ dependence in the Dvoretsky theorem, we can gain a deeper understanding of the relationship between the volume of a convex body and its diameter, and develop new techniques for estimating the volume of convex bodies.