I have a question about a property of Lipschitz domain.



> 
> Let $D \subset \mathbb{R}^d$ be a bounded domain (connected open subset ). $D$ is called **a
>  bounded Lipschitz domain** if there exist positive constants $\delta$,
>  $M$ such that for each $x\_0 \in \partial \Omega$ there exist a
>  neighborhood $U\_{x\_0}$ of $x\_0$, local coordinates $y=(y',y\_d) \in
>  \mathbb{R}^{d-1} \times \mathbb{R}$, with $y=0$ at $x\_0$, and a 
>  Lipschitz continous function $f\_{x\_0}:\mathbb{R}^{d-1} \to
>  \mathbb{R}$, such that \begin{equation\*} D \cap U\_{x\_0}=\{(y',y\_N) \in
>  D \cap U\_{x\_0} \mid y' \in B(0,\delta),\, y\_N>f(y') \},\quad
>  \text{Lip}(f) \le M, \end{equation\*} where we define
>  $\text{Lip}(f)=\inf \{L \ge 0 \mid |f(x)-f(y)| \le L|x-y|,\, x,y \in
> B(0,\delta) \}$.
> 
> 
> 


In the above definition, $B(0,\delta)$ denotes the open ball in $\mathbb{R}^d$ centered at the origin with radius $1$.


**My question**


We denote by $m$ the Lebesgue measure on $D$.


Let $D \subset \mathbb{R}^d$ be a bounded Lipschitz domain. Then, can we show the following?


There exists a positive constant $C \ge 1$ such that 
$$C^{-1}r^d \le m(\bar{D} \cap B(x,r)) \le Cr^d$$
for any $x \in \bar{D}$, $r \in (0,\text{diam}(D)]$.


It is clear that $m(\bar{D} \cap B(x,r)) \le Cr^d$ for any $x \in \bar{D}$, $r \in (0,\text{diam}(D)]$.


If you know a proof or references, please let me know.



A property of Lipschitz domains
This is probably somewhat over-the-top, but anyway: The nice paper [1] by Hajlasz, Koskela and Tuominen says that your desired inequality is true for Sobolev extension domains, so for domains $D$ for which there exists a continuous linear operator $E \colon W^{1,p}(D) \to W^{1,p}(\mathbb{R}^n)$ such that $(Eu)\_{\restriction D} = u$. It is a classical result that a Lipschitz domain is such an extension domain, e.g. Theorem 1.4.3.1 in Grisvard [2].


[1] *Hajłasz, Piotr; Koskela, Pekka; Tuominen, Heli*, [**Sobolev embeddings, extensions and measure density condition**](http://dx.doi.org/10.1016/j.jfa.2007.11.020), J. Funct. Anal. 254, No. 5, 1217-1234 (2008). [ZBL1136.46029](https://zbmath.org/?q=an:1136.46029). 


[2] *Grisvard, Pierre*, Elliptic problems in nonsmooth domains, Classics in Applied Mathematics 69. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-611972-02-3/pbk). xx, 410 p. (2011). [ZBL1231.35002](https://zbmath.org/?q=an:1231.35002).


