I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner.
Let $\Omega\subset\Bbb R^d$ be an open bounded (may be connected just to make it simpler)
For $\delta>0$ small enough we define the shrunken version of $\Omega$ as by $$\Omega_\delta= \{x\in \Omega~: \operatorname{dist}(x,\partial \Omega)>\delta\}$$
Basically, for $\delta$ small enough, $\Omega$ and $\Omega_\delta$ have a similar shape. For instance if $\Omega= B(0,1)$ is the unit ball then, $\Omega_\delta=B(0,\delta) =\delta B(0,1)$ is the ball centered at 0 with radius $\delta$.
Definition An open set $\Omega$ is said to satisfy the Volum Density Condition(VDC) if there exists a constant $\kappa>0$ such that for all $x\in \partial \Omega$ and $r\in (0,1)$ $$|\Omega\cap B(x,r)|\geq \kappa r^d.$$
An open set $\Omega$ is said to be of class $C^k$ if for every $x\in \partial \Omega$ there exists $r>0$ and a mapping $\gamma: \Bbb R^{d-1}\to\Bbb R$ such that
$$ \Omega\cap B(x,r)= \{x=(x',x_d)\in B(x,r)~: x_d>\gamma(x')\}$$
A natural question would be: Does $\Omega_\delta$ inherit the regularity properties of $\Omega$? Precisely,
1)If $\Omega$ is $C^k$ do we have that $\Omega_\delta$ is also $C^k$?
2) If $\Omega$ satisfies the Volum Density Condition(VDC) does $\Omega_\delta$ also satisfies the Volum Density Condition(VDC)?
