The Gaussian Free-Energy Pair is Uniformly Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector, with greatest variance $c_{\max}$, let $a\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian free-energy pair with variances $c$ and temperature $a$, a penalty pair by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §pair. $\lambda$-displacement convexity, closed score along couplings and regular penalised maxima are those of the definitions cited.

1. (Uniform displacement convexity)¶ The pair is $\frac{a}{c_{\max}}$-displacement convex; in particular it is displacement convex.

2. (Closed score)¶ The pair has closed score along couplings.

3. (Regular penalised maxima)¶ The pair has regular penalised maxima.