$\lambda$-Displacement Convexity of a Penalty Pair on the Wasserstein Space

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\lambda\in\mathbb{R}$. For $\mu\in\mathcal{D}_{\Sigma}$ one has $\mu\in\mathcal{D}$, so that $\mathcal{E}(\mu)$ is a real number, and the score $\Sigma(\mu)$ lies in $T_{\mu}$, hence in $L^{2}(\mu;\mathbb{R}^{d})$, both by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i); so for $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$ the displacement pairing $\mathcal{J}(\Sigma(\mu),\pi)$ and the cost $I(\pi)$ are the real numbers fixed there. Optimal couplings are those of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i), and $\tfrac{\lambda}{2}$ is the product of $\lambda$ with the multiplicative inverse of $2$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field.

($\lambda$-displacement convexity)(i) The penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is $\lambda$-displacement convex if

$$\mathcal{E}(\mu)+\mathcal{J}(\Sigma(\mu),\pi)+\tfrac{\lambda}{2}\,I(\pi)\ \le\ \mathcal{E}(\nu)$$

for every $\mu\in\mathcal{D}_{\Sigma}$, every $\nu\in\mathcal{D}$ and every optimal coupling $\pi\in\Pi(\mu,\nu)$. It is displacement convex if it is $0$-displacement convex.