I have heard in a few talks that the unit norm ball on the cotangent space of Teichmüller space is convex. I am looking for a reference where that statement would be proved.
I am also looking for a reference to a theorem which is, I believe, much stronger. I have been told that a point in Teichmüller space is determined by the unit norm ball on the tangent space (Beltrami differentials) and I would also like to read about that.
You can find a proof of the second result in Royden's paper "Automorphisms and isometries of Teichmüller space". The relevant theorem is Theorem 1, where Royden shows that for the Teichmüller space $\mathcal{T}_g$, if $\dim_{\mathbb{C}} \mathcal{T}_g > 1$ then for every $X \in \mathcal{T}_g$ the normed space $(Q(X),\| \cdot \|)$ determines $X$ up to isomorphism.
As for the first result, I believe is just true by general functional analysis: balls in normed vector spaces are convex.