Let $\Sigma$ be a 2-sphere with a (smooth) Riemannian metric $g$ of nonnegative curvature. Let $\mathcal{C}$ be a simple closed smooth curve on $\Sigma$. Then $\mathcal{C}$ splits $\Sigma$ into two components $\Sigma^+$, $\Sigma^-$. Set $S(t):=\{p\in\Sigma^+\mid d(\mathcal{C},p)=t\}$ where $d$ is the distance function induced by $g$. If I rememeber correctly, the length $L(t)$ of $S(t)$ is well-defined for almost all $t\geq 0$. 


Is it true that $$|L(t\_1)-L(t\_2)|\leq 2\pi|t\_1-t\_2|$$ 
whenever defined? (At least when $S(t\_1),S(t\_2)$ are non-empty.)



Bishop-Gromov type inequality for Jordan curve on 2-sphere
Cut out a small arc $AB$ of the equatorial circle of the round $S^2$ and replace it by minimizing geodesics connecting $A$ and $B$ to the north pole. Denote the resulting curve $\mathcal C$. Then the curves $S\_t$ south of $\mathcal C$ vary in length wildly for small $t$, violating your bound. 


