= Displacement convexity
{title2=$E(\mu_t)\leq(1-t)E(\mu_0)+tE(\mu_1)-\frac\lambda2t(1-t)W_2(\mu_0,\mu_1)^2$}
An energy has $\lambda$-displacement convexity when the displayed inequality holds along every <displacement interpolation> between endpoints in its domain. The case $\lambda=0$ is displacement convexity; $\lambda>0$ gives uniform convexity along these <metric geodesics>. This is distinct from <convexity> along ordinary linear mixtures of <probability measures>. On subintervals, the inequality is equivalent to <convexity> of $t\mapsto E(\mu_t)-\lambda W_2(\mu_0,\mu_1)^2t^2/2$; if the energy is twice differentiable, it is equivalent to the corresponding <second derivative> lower bound. Without that regularity the lower bound is interpreted using <distributions>.
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