Is the barycenter of a convex curve in $\mathbb R^2$ Lipschitz with respect to the Hausdorff distance?

Question

For a curve $C$, its barycenter is $$\text{Bar}(C) = \frac{1}{\text{length}(C)}\int\limits_C x d \mathcal H^1(x).$$ Does there exist a constant $L$ such that for $C_1,C_2$ convex curves in the plane, $$|\text{Bar}(C_1) - \text{Bar}(C_2)| \leq L d_H(C_1,C_2),$$ where $d_H$ is the Hausdorff distance?