Local Bounds for the Delta-Envelope and Attained Maxima on Closed Balls, for a Wasserstein-Coercive Penalty Pair

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-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\delta\in\mathbb{R}$ be positive, let $\hat{\mu}\in\mathcal{D}$, and let $r,c\in\mathbb{R}$ with $r$ positive. Upper semicontinuity on $\mathcal{D}$ relative to $\mathcal{D}$ is understood 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; that a function is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, and the $\delta$-envelope $u^{-}_{\delta}$ of such a function $u$, a function on $\mathcal{D}$, are those of the definitions cited. Put

which contains $\hat{\mu}$ because $W_{2}(\hat{\mu},\hat{\mu})=0$, $W_{2}$ being a metric by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric.

1. (Envelope bound)¶ Let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and satisfy $u(\nu)\le c$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\nu,\hat{\mu})<r$. Then

2. (Attained maximum)¶ Let $g:\mathcal{D}\to\mathbb{R}$ be upper semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ and satisfy $g(\nu)\le c-\delta\,\mathcal{E}(\nu)$ for every $\nu\in K$. Then there is $\nu^{*}\in K$ with $g(\nu)\le g(\nu^{*})$ for every $\nu\in K$.