SCHWEITZER 2014 Problem 9

Let $\rho:\mathbb{R}^n\to \mathbb{R}$ , $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$ , and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula $$\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.$$ Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$ .