Let $K_1$, $K_2$ be two convex compact sets in $\mathbb{R}^d$, and $p_1,p_2$ be their barycenters. Is it true that the distance between $p_1$ and $p_2$ does not exceed a Hausdorff distance between $K_1$ and $K_2$? If not, maybe there is some weaker estimate, say, uniform continuity of the map (convex body)$\rightarrow$ (its barycenter) for bodies inside, say, unit ball?