A paper on Approximation Theory.

In summary: Your name]In summary, the paper discusses a specific property of Lipschitz domains called the "cone property". This property states that for any point on the boundary of a Lipschitz domain, there exists a cone with a specific head angle that is contained within the domain. The property (i) mentioned in the paper is a variation of this cone property, and its value depends on the dimension and radius of the domain. The author does not explicitly state the value of $\delta$, but it can be inferred from the proof of Lemma 2.1. The inequality $\sin(\alpha/2)\ge 1/R$ is derived from the fact that the head angle of the cone is at least $2\arcs
  • #1
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I asked my question in overflow, so far with no answers.

Perhaps here, I'll get an answer.

https://mathoverflow.net/questions/282048/a-lemma-on-convex-domain-which-is-a-lipschitz-domain

[admin edit: Below is the actual question posted, so our community doesn't have to follow multiple links:]

[box=yellow]I am reading the following paper:

Whitney Estimates for Convex Domains with Applications to Multivariate Piecewise Polynomial Approximation

I don't see why does the property (i) of Lipschitz domain is satisfied? (it's listed on the bottom of page 5), what is $\delta:=\delta(d,R)$ explicitly?

I don't understand the last paragraph:

[box=gray]Now, for any $x\in \partial \Omega \cap U_j$, the cone defined by the convex closure of $x\cup B(0,1)$ is contained in the closure of $\Omega$. Any head angle $\alpha$ of this cone satisfies $\sin(\alpha/2)\ge 1/R$ and thus the boundary is $Lip 1$ function with $M:=M(d,R)$.[/box]

Can you elaborate on the two sentences in the quote above? and with the specifics, cause I don't see why the convex closure of $x\cup B(0,1)$ cannot exceed the closure of $\Omega$; I also don't see how did they arrive at the inequality in the last sentence of $\sin(\alpha/2)$, perhaps I am missing something from elementary geometry. why is the boundary $Lip1$?

Thanks.[/box]

Cheers!
 
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  • #2

Thank you for reaching out to our community for help with your question. I am a scientist and I would be happy to assist you with understanding the paper you are reading.

After reviewing the paper and your question, I believe that the author is referring to a specific property of Lipschitz domains, which is the "cone property". This property states that for any point on the boundary of a Lipschitz domain, there exists a cone with a specific head angle that is contained within the domain. This cone is defined by the convex closure of the point and a ball of radius 1 centered at the origin. The property (i) mentioned in the paper is a variation of this cone property, where the head angle of the cone is dependent on the dimension of the domain and its radius.

The author does not explicitly state the value of $\delta$ in the paper, but it can be inferred from the proof of Lemma 2.1. The value of $\delta$ depends on the dimension $d$ and the radius $R$ of the domain, and it is used to define the head angle of the cone. The inequality $\sin(\alpha/2)\ge 1/R$ is derived from the fact that the head angle of the cone is at least $2\arcsin(1/R)$ (as mentioned in the paper), and using elementary geometry, we can show that $\sin(\alpha/2)\ge 1/R$.

As for the convex closure of $x\cup B(0,1)$ not exceeding the closure of $\Omega$, this is also a consequence of the cone property. Since the cone is contained within the domain, its convex closure (which is the smallest convex set containing the cone) must also be contained within the domain. Therefore, the convex closure of $x\cup B(0,1)$ cannot exceed the closure of $\Omega$.

I hope this helps clarify your understanding of the paper and the specific property being discussed. If you have any further questions or need clarification on any other parts of the paper, please do not hesitate to ask. Our community is here to help you.


 

FAQ: A paper on Approximation Theory.

1. What is Approximation Theory?

Approximation Theory is a branch of mathematics that deals with finding approximate solutions to mathematical problems. It involves the use of mathematical tools to represent complex functions or data sets with simpler, more manageable ones.

2. How is Approximation Theory used in scientific research?

Approximation Theory is used in various fields of science, including physics, engineering, and computer science. It is used to model and analyze complex systems and to find approximations for solutions that would otherwise be impossible to obtain.

3. What are some common techniques used in Approximation Theory?

Some common techniques used in Approximation Theory include polynomial interpolation, spline interpolation, least squares approximation, and Fourier analysis. These techniques involve creating simpler representations of functions or data sets by using a finite number of simpler functions.

4. Can Approximation Theory be used for real-world applications?

Yes, Approximation Theory has many real-world applications. It is used in fields such as signal processing, data compression, and image and signal reconstruction. It is also used in scientific research to approximate complex mathematical models and equations.

5. What are the limitations of Approximation Theory?

While Approximation Theory is a powerful tool, it does have limitations. One limitation is that the accuracy of the approximation depends on the choice of the simpler functions used. Another limitation is that it may not always be possible to find an approximation that accurately represents the original function or data set.

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