Let $M \subset \mathbb{R}^n$ be open, bounded and convex and equip $M$ with an unbounded metric $d$ that induces the Euclidean topology. Is there always a map $f : M \to M$ and two constants $C_1 > 0$ and $C_2 < 1$ such that $$C_1d(x,y) \leq d(f(x),f(y)) \leq C_2d(x,y) \quad$$ for all $x,y \in M$?

I tried fixing a point and contracting along straight lines, but I somehow fail to see why this yields such a map...

What if we take an unbounded (simply connected) Riemannian manifold $(M,g)$ with induced metric $d$ instead?

Let $M \subset \mathbb{R}^n$ be open, bounded and convex and equip $M$ with an unbounded metric $d$ that induces the Euclidean topology. Is there always a map $f : M \to M$ and two constants $C_1 > 0$ and $C_2 < 1$ such that $$C_1d(x,y) \leq d(f(x),f(y)) \leq C_2d(x,y) \quad$$ for all $x,y \in M$?

I tried fixing a point and contracting along straight lines, but I somehow fail to see why this yields such a map...

What if we take an unbounded (simply connected) Riemannian manifold $(M,g)$ with induced metric $d$ instead?

Edit: Since it turned out to be wrong: Is it still wrong if we require the map to be proper instead? I.e. is there always a contractive proper map?