Let $n\in\mathbb N$, and let $U:[0,\infty)\to\mathbb R$ be convex, continuous at $0$, and satisfy $U(0)=0$. Define $h:(0,\infty)\to\mathbb R$ by
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\begin{align*}
h(r)=r^n U(r^{-n}).
\end{align*}
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Assume that $h$ is convex and nonincreasing on $(0,\infty)$.
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For $\mu\in\mathcal P_2(\mathbb R^n)$, define $\mathcal U:\mathcal P_2(\mathbb R^n)\to(-\infty,+\infty]$ as follows. If $\mu$ is absolutely continuous with respect to $\mathcal L^n$, write $\mu=\rho\,d\mathcal L^n$ for its Borel density and set
whenever this extended integral is well-defined. In all other cases, including every non-absolutely-continuous $\mu$, set $\mathcal U[\mu]=+\infty$.
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Then $\mathcal U$ is displacement convex in the following precise sense. For every $\mu_0,\mu_1\in\mathcal P_2(\mathbb R^n)$ there exists a constant-speed $W_2$-geodesic $(\mu_t)_{t\in[0,1]}$ from $\mu_0$ to $\mu_1$ such that, for every $t\in[0,1]$,
with the usual extended-real convention that the right-hand side is $+\infty$ if at least one endpoint energy is $+\infty$ and its coefficient is positive.
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Assume in addition that the following lower-semicontinuity hypothesis holds for $U$: whenever $(\mu_k)_{k\in\mathbb N}\subset\mathcal P_2(\mathbb R^n)$ converges narrowly to $\mu\in\mathcal P_2(\mathbb R^n)$ and has uniformly bounded second moments, one has
For example, this hypothesis is satisfied under the standard hypotheses of the integral-functional lower-semicontinuity theorem for convex superlinear densities, such as an appropriate affine lower control of the negative part together with superlinear growth at infinity.