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

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use. By Basic Properties of a Wasserstein-Coercive Penalty Pair (iii) the penalty $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$, and by Basic Properties of a Wasserstein-Coercive Penalty Pair (ii) there is $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$. The symmetry and the triangle inequality of $W_{2}$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space (i), The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space (iii)) are used without further mention.

Claim 1. Put $O=\{\nu\in\mathcal{P}_{2}(\mathbb{R}^{d}):W_{2}(\nu,\hat{\mu})<r\}$, which is open in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ by Open Ball in a Metric Space is Open, and $S'=\mathcal{D}\cap O$, which contains $\hat{\mu}$ and is therefore nonempty. By The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair (i) the function $u-\delta\mathcal{E}$ on $\mathcal{D}$ is bounded above near each point of $\mathcal{D}$ and $u^{-}_{\delta}=(u-\delta\mathcal{E})^{*}$. By Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set (i) its restriction $(u-\delta\mathcal{E})|_{S'}$ is bounded above near each point of $S'$, and by Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set (ii)

$$\bigl((u-\delta\mathcal{E})|_{S'}\bigr)^{*}(\nu)=u^{-}_{\delta}(\nu)\qquad(\nu\in S').$$

Let $h:S'\to\mathbb{R}$ be $h(\nu)=c-\delta\,\mathcal{E}(\nu)$. We show that $h$ is upper semicontinuous on $S'$. Let $\nu_{0}\in S'$ and let $\varepsilon\in\mathbb{R}$ be positive; then $\varepsilon\delta^{-1}$ is positive by claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field. By Lower Semicontinuous Function on a Subset of a Metric Space applied to $\mathcal{E}$ at $\nu_{0}$ with $\varepsilon\delta^{-1}$, there is a positive $\theta$ with $\mathcal{E}(\nu_{0})-\varepsilon\delta^{-1}<\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$ with $W_{2}(\nu_{0},\nu)<\theta$. For such $\nu$ lying in $S'$, multiplying by the positive $\delta$ (claim 10 of Elementary Order Arithmetic in an Ordered Field) gives $\delta\,\mathcal{E}(\nu_{0})-\varepsilon<\delta\,\mathcal{E}(\nu)$, and hence, by claim 1 of that lemma applied twice, $c-\delta\,\mathcal{E}(\nu)<c-\delta\,\mathcal{E}(\nu_{0})+\varepsilon$, that is $h(\nu)<h(\nu_{0})+\varepsilon$. So $h$ is upper semicontinuous at $\nu_{0}$ relative to $S'$ by Upper Semicontinuous Function on a Subset of a Metric Space.

For $\nu\in S'$ we have $u(\nu)\le c$ by hypothesis, so $u(\nu)-\delta\,\mathcal{E}(\nu)\le h(\nu)$ by the compatibility of the order with addition. Properties of the Upper Semicontinuous Envelope (iii), applied on $S'$ to $(u-\delta\mathcal{E})|_{S'}$ and $h$, gives $\bigl((u-\delta\mathcal{E})|_{S'}\bigr)^{*}(\nu)\le h(\nu)$ for $\nu\in S'$. With the identity displayed above, $u^{-}_{\delta}(\nu)\le c-\delta\,\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$ with $W_{2}(\nu,\hat{\mu})<r$.

Claim 2. For $\nu\in K$ we have $\delta e_{0}\le\delta\,\mathcal{E}(\nu)$ by claim 5 of Elementary Arithmetic in an Ordered Field, hence $g(\nu)\le c-\delta\,\mathcal{E}(\nu)\le c-\delta e_{0}$. The set $\{g(\nu):\nu\in K\}$ is therefore nonempty, since $\hat{\mu}\in K$, and bounded above by $c-\delta e_{0}$; let $s$ be its least upper bound, which exists by The Real Numbers: Standing Notation and Background (iv). Then $g(\hat{\mu})\le s$.

By Existence of a Sequence of Positive Real Numbers with Limit Zero there is a sequence $(h_{n})_{n\in\mathbb{N}}$ of positive real numbers with limit $0$; replacing $h_{n}$ by $\min\{h_{n},1\}$ we may assume $h_{n}\le1$ for every $n$, the new sequence being positive by claim 9 of Elementary Order Arithmetic in an Ordered Field and still having limit $0$ by claim 2 of Order Properties of Limits of Real Sequences, squeezed between the constant sequence $0$ and $(h_{n})$. For each $n\in\mathbb{N}$, claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$, applied with $\varepsilon=h_{n}$, provides $\nu_{n}\in K$ with

$$s-h_{n}<g(\nu_{n})\le s .$$

Then $g(\hat{\mu})-1\le s-h_{n}<g(\nu_{n})$, so, using $g(\nu_{n})\le c-\delta\,\mathcal{E}(\nu_{n})$ and claim 1 of Elementary Order Arithmetic in an Ordered Field, $\delta\,\mathcal{E}(\nu_{n})<c-g(\hat{\mu})+1$, and multiplying by $\delta^{-1}$ (claims 7 and 10 of that lemma),

$$\mathcal{E}(\nu_{n})\le c'\qquad\text{with}\qquad c'=\delta^{-1}\bigl(c-g(\hat{\mu})+1\bigr)\qquad(n\in\mathbb{N}).$$

By Wasserstein-Coercive Penalty Pairs (i) the set $\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c'\}$ is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, so by Sequentially Compact Subset of a Metric Space there are $\nu^{*}$ in that set, in particular $\nu^{*}\in\mathcal{D}$, and a strictly increasing sequence $(n_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ such that $(\nu_{n_{k}})_{k\in\mathbb{N}}$ converges to $\nu^{*}$.

$\nu^{*}$ lies in $K$. Let $\varepsilon\in\mathbb{R}$ be positive. By Convergent Sequence in a Metric Space there is $k$ with $W_{2}(\nu_{n_{k}},\nu^{*})<\varepsilon$, and then $W_{2}(\nu^{*},\hat{\mu})\le W_{2}(\nu^{*},\nu_{n_{k}})+W_{2}(\nu_{n_{k}},\hat{\mu})\le\varepsilon+r$, since $\nu_{n_{k}}\in K$. As $\varepsilon$ was arbitrary, $W_{2}(\nu^{*},\hat{\mu})\le r$ by Comparison of Real Numbers with Arbitrary Positive Slack (i), so $\nu^{*}\in K$ and $g(\nu^{*})\le s$.

$\nu^{*}$ is a maximiser. The real sequence $(h_{n})$ converges to $0$ in the metric space $(\mathbb{R},d_{\mathbb{R}})$, $d_{\mathbb{R}}(x,y)=|x-y|$ being the metric of The Absolute Value Metric on the Real Numbers, so that convergence in $(\mathbb{R},d_{\mathbb{R}})$ is exactly the condition of Limit of a Sequence of Real Numbers; by A Subsequence of a Convergent Sequence Has the Same Limit the subsequence $(h_{n_{k}})_{k\in\mathbb{N}}$ converges to $0$ as well. Let $\varepsilon\in\mathbb{R}$ be positive. By Upper Semicontinuous Function on a Subset of a Metric Space there is a positive $\theta$ with $g(\nu)<g(\nu^{*})+\varepsilon$ for every $\nu\in\mathcal{D}$ with $W_{2}(\nu^{*},\nu)<\theta$. Choose $k$ so large that $W_{2}(\nu_{n_{k}},\nu^{*})<\theta$ and $h_{n_{k}}<\varepsilon$, which is possible because both conditions hold from some index on and the larger of the two indices serves. Then

$$s-\varepsilon<s-h_{n_{k}}<g(\nu_{n_{k}})<g(\nu^{*})+\varepsilon ,$$

so $s\le g(\nu^{*})+2\varepsilon$. As $\varepsilon$ was arbitrary, $s\le g(\nu^{*})$ by Comparison of Real Numbers with Arbitrary Positive Slack (i) (applied with the positive numbers $2\varepsilon$, which exhaust the positive reals by claim 8 of Elementary Order Arithmetic in an Ordered Field). Together with $g(\nu^{*})\le s$ this gives $g(\nu^{*})=s$, and so $g(\nu)\le s=g(\nu^{*})$ for every $\nu\in K$.