Assume $\Omega$ is a $C^1$ bounded domain of $\mathbb{R}^d$, $d \geq 2$. For $(x,y) \in (\partial \Omega)^2$, we say $x \sim y$ if $n_x \cdot (y-x) > 0$, $n_y \cdot (x-y) > 0$, and $(tx+(1-t)y) \in \Omega$ for all $t \in (0,1)$.
Is it true that for any $(x,y) \in (\partial \Omega)^2$ such that $x \sim y$, one can find $\epsilon_1 > 0, \epsilon_2 > 0$ such that $(B(x,\epsilon_1) \cap \partial \Omega) \sim (B(y,\epsilon_2) \cap \partial \Omega)$ (in the sense that for any $a \in (B(x,\epsilon_1) \cap \partial \Omega)$, $b \in B(y,\epsilon_2) \cap \partial \Omega)$, $a \sim b$)?
From the $C^1$ property of the domain it seems clear that for all $(x,y) \in \partial \Omega^2$ with $x \sim y$, one can find $\epsilon > 0$ such that $x \sim B(y,\epsilon) \cap \partial \Omega$ (and I have a proof for it). However the stronger result that I ask, although I don't see any reason why this should not hold, puzzles me a lot more.
Any idea whether the statement is true ? And how to show it ?