MathDB

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

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