New answers tagged geometric-topology
Consider the infinite binary rooted tree $T$, and consider the tree $T'$ obtained by gluing to $T$ an infinite ray to the root. Then there are quasi-isometric embeddings between $T$ and $T'$ in both directions but no quasi-isometry between them, since their boundary are not homeomorphic (Cantor versus (Cantor plus isolated point)).
The same argument works in the relative case. Consider the integral of $d\alpha$ over the convex surface. Since the boundary is Legendrian, $\alpha$ vanishes identically on it. Thus $d\alpha$ has to change sign.
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