If $\Omega$ is a convex open set in $\mathbb{R} ^n$ and $A\subseteq \Omega$ is compact, then $x+(1-c|x-y|)(z-y)\in\Omega$

Question

If $\Omega$ is a convex open set in $\mathbb{R} ^n$ and $A\subseteq \Omega$ is compact, then $x+(1-c|x-y|)(z-y)\in\Omega$ for some $c >0$ and for all $z\in \Omega$ and $x,y\in A$.
If we pick an open cover $\mathcal U =\{N(p,d(p)\}_{p\in \partial A}$ where $N(p,d(p))=\{q: |q-p|0$ it is clear that $\inf |s-p_i| \ge \delta$ for $s\in \Omega ^c$. What I wanted to achieve with this is to show that for any $x \in A$ there is a neighborhood of a fixed size around $x$ completely contained in $\Omega$ (perhaps this neighborhood should have radius $\delta$) and then prove that $x+(1-c|x-y|)(z-y)$ was a convex combination of some vectors within this neighborhood. However, I have not been able to prove either of those things because I have not shown that $\inf |s-x|>0$, and I do not know how to proceed from that, if I were able to prove it. Any hints are appreciated.