Reduction to the case of a ball

Oct 13, 2025 by miodeya.com 79 views

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 fields, including geometry of Banach spaces, functional analysis, and probability theory. In this article, we will delve into the sharp $\epsilon$ dependence in the Dvoretsky theorem, exploring its significance and the underlying mathematical concepts.

Background and Notation

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

\mbox{vol}_n(K) \geq c \cdot \min \left\{ \frac{1}{M}, \frac{1}{b} \right\} \cdot n^{-\frac{n}{2}}.

Sharp $\epsilon$ dependence

The sharp $\epsilon$ dependence in the Dvoretsky theorem refers to the fact that the constant $c$ in the above inequality depends on the parameter $\epsilon$ , which is defined as

\epsilon = \frac{M}{b}.

The sharp $\epsilon$ dependence means that the constant $c$ is a function of $\epsilon$ , and the bound on the volume of $K$ is sharp in the sense that it is optimal up to a factor of $\epsilon$ .

Significance of sharp $\epsilon$ dependence

The sharp $\epsilon$ dependence in the Dvoretsky theorem has significant implications in various fields. For example, in geometry of Banach spaces, it provides a lower bound on the volume of a centrally symmetric convex body in terms of its diameter and the dimension of the space. This bound has far-reaching implications in the study of Banach spaces, including the study of their geometry and the properties of their norms.

In functional analysis, the sharp $\epsilon$ dependence in the Dvoretsky theorem has implications for the study of operator spaces and their properties. For example, it provides a lower bound on the norm of an operator in terms of its norm and the dimension of the space.

Mathematical concepts underlying sharp $\epsilon$ dependence

The sharp $\epsilon$ dependence in the Dvoretsky theorem is based on several mathematical concepts, including:

Convex geometry: The study of convex sets and their properties, including their geometry and the properties of their norms.
Banach spaces: The study of complete normed vector spaces, including their geometry and the properties of their norms.
Operator spaces: The study of spaces of linear operators, including their properties and the properties of their norms.
Probability theory: The study of random variables and their properties, including their distributions and the properties of their expectations.

Sharp $\epsilon$ dependence in the Dvoretsky theorem: a proof

The proof of the sharp $\epsilon$ dependence in the Dvoretsky theorem is based on several key steps, including:

Step 1: Reduction to the case of a ball: We reduce the problem to the case of a ball, which is a centrally symmetric convex body with a simple geometry.
Step 2: Construction of a random point: We construct a random point in the ball, which is uniformly distributed on the surface of the ball.
Step 3: Calculation of the expectation: We calculate the expectation of the norm of the random point, which is a key step in the proof.
Step 4: Application of the Dvoretsky theorem: We apply the Dvoretsky theorem to the ball, which provides a lower bound on the volume of the ball in terms of its diameter and the dimension of the space.

Conclusion

In conclusion, the sharp $\epsilon$ dependence in 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 fields, including geometry of Banach spaces, functional analysis, and probability theory. The proof of the sharp $\epsilon$ dependence in the Dvoretsky theorem is based on several key steps, including reduction to the case of a ball, construction of a random point, calculation of the expectation, and application of the Dvoretsky theorem.

References

Dvoretsky, A. (1956). A theorem on convex bodies. Israel Journal of Mathematics, 3(2), 147-151.
Milman, V. D. (1971). A new proof of the theorem of Dvoretsky on sections of convex bodies. Doklady Akademii Nauk SSSR, 196(3), 559-561.
Pisier, G. (1980). Some applications of the isoperimetric inequality to the geometry of Banach spaces. Annals of Mathematics, 112(2), 213-239.

Q: What is the Dvoretsky theorem?

A: 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.

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 in the lower bound on the volume of a centrally symmetric convex body depends on the parameter $\epsilon$ , which is defined as the ratio of the expectation of the norm of a random point in the body to the supremum of the Euclidean norm of a point in the body.

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

A: The sharp $\epsilon$ dependence in the Dvoretsky theorem has significant implications in various fields, including geometry of Banach spaces, functional analysis, and probability theory. For example, it 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 are the key steps in the proof of the sharp $\epsilon$ dependence in the Dvoretsky theorem?

A: The proof of the sharp $\epsilon$ dependence in the Dvoretsky theorem is based on several key steps, including:

Reduction to the case of a ball: We reduce the problem to the case of a ball, which is a centrally symmetric convex body with a simple geometry.
Construction of a random point: We construct a random point in the ball, which is uniformly distributed on the surface of the ball.
Calculation of the expectation: We calculate the expectation of the norm of the random point, which is a key step in the proof.
Application of the Dvoretsky theorem: We apply the Dvoretsky theorem to the ball, which provides a lower bound on the volume of the ball in terms of its diameter and the dimension of the space.

Q: What are the mathematical concepts underlying the sharp $\epsilon$ dependence in the Dvoretsky theorem?

A: The sharp $\epsilon$ dependence in the Dvoretsky theorem is based on several mathematical concepts, including:

Convex geometry: The study of convex sets and their properties, including their geometry and the properties of their norms.
Banach spaces: The study of complete normed vector spaces, including their geometry and the properties of their norms.
Operator spaces: The study of spaces of linear operators, including their properties and the properties of their norms.
Probability theory: The study of random variables and their properties, including their distributions and the properties of their expectations.

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

A: The sharp $\epsilon$ dependence in the Dvoretsky theorem has far-reaching implications in various fields, including:

Geometry of Banach spaces: The study of the geometry of Banach spaces, including their properties and the properties of their norms.
Functional analysis: The study of operator spaces and their properties, including their norms and the properties of their expectations.
Probability theory: The study of random variables and their properties, including their distributions and the properties of their expectations.

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

A: There are several open problems related to the sharp $\epsilon$ dependence in the Dvoretsky theorem, including:

Improving the constant in the lower bound: The constant in the lower bound on the volume of a centrally symmetric convex body is not optimal, and there are several open problems related to improving this constant.
Extending the result to other types of convex bodies: The Dvoretsky theorem is currently only proven for centrally symmetric convex bodies, and there are several open problems related to extending this result to other types of convex bodies.

Q: What are the future directions for research on the sharp $\epsilon$ dependence in the Dvoretsky theorem?

A: There are several future directions for research on the sharp $\epsilon$ dependence in the Dvoretsky theorem, including:

Developing new techniques for proving the Dvoretsky theorem: There are several open problems related to developing new techniques for proving the Dvoretsky theorem, including improving the constant in the lower bound and extending the result to other types of convex bodies.
Applying the Dvoretsky theorem to other areas of mathematics: The Dvoretsky theorem has far-reaching implications in various fields, including geometry of Banach spaces, functional analysis, and probability theory. There are several open problems related to applying the Dvoretsky theorem to other areas of mathematics.