= Interaction energy
{title2=$\mathcal W[\mu]=\frac12\iint W(x-y)\,d\mu(x)\,d\mu(y)$}
The factor $1/2$ counts unordered pairs once. For a symmetric <interaction potential>, this energy is unchanged by translating the entire <probability measure>. It therefore cannot have positive uniform <displacement convexity> on the unrestricted space containing translations. If $D^2W\geq\lambda I$, integrate the pointwise strong <convexity> inequality along an optimal <transport plan> and its independent copy. The <variance identity for independent copies> then gives $\lambda$-<displacement convexity> on every class with fixed <barycenter of a measure>, whenever the integrals are finite.
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