Assume that $g$ is the Euclidean metric on $\mathbb{R}^2$ and let $g'$ be another smooth metric on $\mathbb{R}^2$ such that there are constants $a,b>0$ such that for every $z \in \mathbb{R}^2$ and $v \in T_z (\mathbb{R}^2) \simeq \mathbb{R}^2$ we have $$ a \langle v,v\rangle_{g} \leq \langle v,v \rangle_{g'} \leq b\langle v,v\rangle_{g}. $$
In such a nice situation, what can we say about the boundary of the balls in $M = (\mathbb{R}^2,g')$? More specifically, can we say that the boundary of any ball in $M$ is a smooth (or piecewise smooth) curve?
Thanks!