Deformation cone of Tesler polytopes

For $\mathit{a}\in {\mathbb{R}}_{\ge 0}^n$, the Tesler polytope ${\text{tes}}_n(\mathit{a})$ is the set of upper triangular matrices with non-negative entries whose hook sum vector is $\text{ba}$. We first give a different proof of the known fact that for every fixed ${\mathit{a}}_0\in {\mathbb{R}}_{>0}^n$, all the Tesler polytopes ${\text{tes}}_n(\mathit{a})$ are deformations of ${\text{tes}}_n({\mathit{a}}_0)$. We then calculate the deformation cone of ${\text{tes}}_n({\mathit{a}}_0)$. In the process, we also show that any deformation of ${\text{tes}}_n({\mathit{a}}_0)$ is a translation of a Tesler polytope. Lastly, we consider a larger family of polytopes called flow polytopes which contains the family of Tesler polytopes and give a characterization on which flow polytopes are deformations of ${\text{tes}}_n({\mathit{a}}_0)$.