Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-closed penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, and for $c\in\mathbb{R}$ write $\mathcal{D}_{c}=\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$. Lower semicontinuity on $\mathcal{D}$ is taken relative to $\mathcal{D}$ in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions; the restriction of $W_{2}$ to $\mathcal{D}_{c}\times\mathcal{D}_{c}$ is again written $W_{2}$, and completeness is that of that definition. Then the following hold.

1. (The penalty is bounded below)¶ There is $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$.

2. (The penalty is lower semicontinuous)¶ $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$.

3. (Complete sublevel sets)¶ For every $c\in\mathbb{R}$, $(\mathcal{D}_{c},W_{2})$ is a metric space, and it is complete.

4. (Bounded distances on sublevel sets)¶ Let $c,B\in\mathbb{R}$ satisfy $M_{2}(\mu)\le B$ for every $\mu\in\mathcal{D}_{c}$, as provided by Wasserstein-Closed Penalty Pairs §bounded. Then $W_{2}(\mu,\nu)^{2}\le4B$ for all $\mu,\nu\in\mathcal{D}_{c}$.

5. (Coercive pairs are Wasserstein-closed)¶ Let $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ be a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, not assumed to be Wasserstein-closed. Then $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ is Wasserstein-closed.