A paper on Approximation Theory.

[Alone]

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!

 

[jvicens]

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.