¹

1 Introduction, main results and state of the art↩︎

1.1 Motivation, the classical Brøndsted-Rockafellar theorem and summary of results↩︎

Any measurement comes with finite precision, and the true state of a system can only be observed up to a tolerance. For this reason, approximate solutions are often as good as, and frequently indistinguishable, from true ones. Since mathematical laws governing physical systems are typically formulated in variational form, it is natural to consider almost critical points. The Brøndsted–Rockafellar theorem occupies a central place in variational analysis. It establishes that every approximate minimizer of a convex functional can be lifted to a nearby point admitting a subgradient of small norm—that is, an almost critical point.

::: {#thmBR .theorem} Theorem 1 (Brøndsted–Rockafellar [1]). Let \(X\) be a real Banach space and \(\Phi:X\to \mathbb{R}\cup\{+\infty\}\) a proper convex lower semicontinuous functional bounded from below. Let \(u\in X\) and \(\varepsilon>0\) be such that \[\Phi(u)\le \inf_{v\in X}\Phi(v)+\varepsilon.\] Then, for any \(\lambda>0\), there exist \(\hat{u}\in X\) and \(\xi\in\partial \Phi(\hat{u})\) satisfying \[\begin{align} (a)\,\,\, \|u-\hat{u}\|\le \lambda \qquad (b)\,\,\, \Phi(\hat{u})\le \Phi(u) \qquad (c)\,\,\, \|\xi\|\le \frac{\varepsilon}{\lambda}. \end{align}\]* :::

One of the main drawbacks of this result is that it is not constructive; its original proof relied on an order argument via Zorn’s lemma, adapted from the proof of the celebrated Bishop–Phelps lemma [2]. Modern proofs use Ekeland’s variational principle [3], which does not require the axiom of choice. Recently, in [4], a constructive version of the theorem was established in Hilbert spaces. The result provides the desired point by means of the resolvent of the subdifferential. The proof relies on what is now referred to as Carlier’s inequality [5]–[7], an estimate relating the Fenchel–Young gap to the squared resolvent residual. The construction in [4] yields that the choices \[\begin{align} \label{constructions} \hat{u} = \Big(\mathrm{id}+\frac{\lambda^2}{\varepsilon}\,\partial\Phi\Big)^{-1}(u) \qquad\text{and}\qquad \xi = \frac{\varepsilon}{\lambda^2}\,(u-\hat{u}) \end{align}\tag{1}\] satisfy items \((a)\)-\((c)\) in the theorem stated above. An advantage of this explicit construction is that it enables more refined arguments in applications of the theorem. It also fits naturally within the frameworks of constructive mathematics and computable analysis. The Brøndsted–Rockafellar theorem has been extended in many directions, including operator-theoretic [8]–[11] and convex-analytic [12]–[17]. But so far only [4] has given an explicit statement where a construction is provided; a similar construction was used in the proof of [16]. This paper is devoted to a constructive extension of the Brøndsted–Rockafellar theorem to geodesic metric spaces. Such spaces include, for instance, normed linear spaces endowed with the distance induced by the norm, the hyperbolic space equipped with its Riemannian metric (and, more generally, \(\operatorname{CAT}(0)\) spaces), and the space of probability measures with finite second moment on the Euclidean space endowed with the Wasserstein distance. The extension to this latter space is particularly significant, since a notion of subdifferential is available; see [18] and [19]. The contributions of this paper can be summarized as follows.

• Extension to geodesic spaces. We provide two versions of the Brøndsted–Rockafellar theorem. The first one comes in the form of Ekeland’s variational principle (Theorem 1 below), while the second one (Theorem 7 below) is more faithful to the original theorem, replacing the bound on the subdifferential by a bound on the metric slope. The resolvent construction in (1 ) is replaced by proximal points (in Hilbert spaces there is no distinction between resolvent and proximal points). We circumvent Carlier’s inequality, which is intrinsically related to the linear structure due to the use of the Legendre-Fenchel transform, through purely metric arguments. We give some pertinent examples illustrating the applicability of the results.

• Caristi fixed point theorem. As an application of Theorem 1, we recover a metric version of the Caristi fixed point theorem for geodesically convex functionals. The result (stated in Proposition 10 below) is constructive and relies on the existence of proximal points.

• A result on global error bounds. We show (in Proposition 11 below) that if a geodesically convex functional satisfies an error bound controlling the distance to the minimizer set by a power of the metric slope, then it necessarily satisfies a quantitative growth estimate. This extends to geodesic metric spaces the classical result, known for convex functionals in Banach spaces, that the metric sub-regularity of the subdifferential implies a polynomial growth condition at the set of minimizers.

• A construction in the Wasserstein space. In Section 3, we provide more refined results than the general theorems in this section for the particular case of the space of probability measures with the Wasserstein distance. In Theorem 12, we obtain an analogue of the construction in (1 ), formulated in terms of optimal transport maps and the minimizing movement scheme. Furthermore, in Theorem 14, we establish the equivalence between the metric sub-regularity of the subdifferential and the so-called entropy–transportation inequality (also referred to as the Talagrand inequality).

Below in this section we present the main results; their proofs are given in the next section, and the results specific to the Wasserstein space are collected in Section 3.

1.2 The theorem in geodesic spaces and some remarks↩︎

Throughout this subsection, \(\big(X, d\big)\) denotes a geodesic space; that is, a metric space such that for every \(u,v\in X\) there exists a constant-speed² geodesic \(\gamma:[0,1]\to X\) such that \(\gamma(0)=u\) and \(\gamma(1) = v\). Additionally, we consider a proper functional \(\Phi \colon X \to \mathbb{R} \cup \{+\infty\}\) bounded from below such that \[\begin{align} \label{geocon} \Phi(\gamma(t)) \leq (1-t)\,\Phi(\gamma(0)) + t\,\Phi(\gamma(1))\quad \forall t\in[0,1] \end{align}\tag{2}\] for every constant-speed geodesic \(\gamma:[0,1]\to X\). Condition (2 ) is referred to as geodesic convexity of \(\Phi\). We fix numbers \(p,q\in (1,+\infty)\) such that \(p^{-1} + q^{-1} = 1\).

1.2.1 The theorem in variational form↩︎

Consider the set-valued mapping \(\operatorname{prox_{\Phi}}:X\rightrightarrows X\) given by \[\operatorname{prox_{\Phi}}(u):= \mathop{\mathrm{arg\,min}}_{v\in X}\big\{\Phi(v) + \frac{1}{p} d(v,u)^p\big\}.\] This operator is usually referred to as the proximal mapping, and in some contexts it can be interpreted as a resolvent operator. The main result of this subsection establishes that the proximal mapping naturally yields points satisfying the conditions of Ekeland’s variational principle.

We now make some remarks.

1.2.2 The theorem with the metric slope↩︎

The metric slope of \(\Phi\) at a point \(u\in X\) is given by \[\begin{align} \label{mesl} |\partial \Phi|(u) := \max\left\{\limsup_{v\longrightarrow u}\frac{\Phi(u)-\Phi(v)}{d(u,v)},0\right\}. \end{align}\tag{3}\] Quantity 3 is always nonnegative, and vanishes exactly at the minimizers of \(\Phi\). If \(X\) is a real normed space, then the metric slope coincides with the distance of the convex subdifferential to zero, i.e., \(|\partial \Phi|(u) =\operatorname{dist}\big(0,\partial \Phi(u)\big)\) for all \(u\in X\), where \(\partial\Phi:X\rightrightarrows X^*\) denotes the convex differential of \(\Phi\); see [20]. This identity is one of the reasons for the term metric slope. We now give an alternative version of Theorem 1.

We now make a couple of remarks.

1.3 A couple of consequences↩︎

We now give a couple of applications of Theorem 1. As in the previous subsection, we consider a geodesic space \(\big(X, d\big)\) and a proper geodesically convex functional \(\Phi \colon X \to \mathbb{R} \cup \{+\infty\}\) bounded from below.

1.3.1 The fixed-point theorem of Caristi↩︎

A classical consequence of Ekeland’s variational principle is the so-called Caristi fixed-point theorem [21]. A constructive variant of this result can be recovered for geodesically convex functionals by means of Theorem 1.

This formulation shows that the existence of minimizers of regularized problems, combined with geodesic convexity and condition ?? , is sufficient to guarantee fixed-point properties, even without completeness or lower semicontinuity.

1.3.2 A necessary condition for metric sub-regularity↩︎

Metric sub-regularity is a fundamental stability property of set-valued mappings; it relates the distance to the solution set with a suitable residual measure; see [22]–[25] for detailed references. This property for convex subdifferentials was first characterized in [26] through growth conditions via Ekeland’s variational principle. Using the metric slope, the global sub-regularity of the subdifferential may be interpreted as a quantitative relation between the slope of a functional and its deviation from the minimum value; see [27]. The following result provides a necessary condition for the sub-regularity property in terms of an error bound over the domain of the proximal mapping.

The growth condition (?? ) is a well-known error bound; see [20], [27], [28]. In Wasserstein spaces, it is common to refer to conditions such as (?? ) as entropy–transport inequalities or Talagrand inequalities; see [29] and [30].

1.4 Some pertinent examples↩︎

We present some relevant examples concerning the applicability of Theorem 1 in situations where the assumptions of the classical Brøndsted–Rockafellar theorem are not satisfied.

1.4.1 Lack of lower semicontinuity↩︎

Let \(H\) be a Hilbert space and \(C\subseteq H\) a nonempty open bounded convex set. Fix \(\xi\in H\) and consider the functional \(\Phi:H\to\mathbb{R}\cup\{+\infty\}\) given by \[\begin{align} \Phi(v)=-\langle\xi,v\rangle + \iota_{sec:C}(v), \end{align}\] where \(\iota_C:H\to\mathbb{R}\cup\{+\infty\}\) denotes the indicator function of \(C\). It is clear from its definition that \(\Phi\) is proper, convex, and bounded from below. Furthermore, it can be seen that

• \(\Phi\) is not lower semicontinuous;

• For any \(u\in H\) and \(t>0\), \[\operatorname{prox}_{t\Phi}(u)= \begin{cases} \{\,u + t^{\,q-1}\|\xi\|^{\,q-2}\xi\,\} & \text{if }\,\, u \in C - t^{\,q-1}\|\xi\|^{\,q-2}\xi, \\[6pt] \varnothing & \text{otherwise.} \end{cases}\]

While the functional is not lower semicontinuous, and the classical Brøndsted-Rockafellar theorem cannot be applied, one can still apply Theorem 1 for specific choices of parameters.

1.4.2 Lack of completeness↩︎

Let \(\Omega\subseteq\mathbb{R}^d\) be a convex bounded open set. Consider the set \[\mathcal{P}_2(\Omega):=\left\{\mu\in\mathcal{P}_2(\mathbb{R}^d):\, \mu(\Omega)=1\right\}\] Preliminaries on the Wasserstein space are given in Subsection 3.1. The tuple \((\mathcal{P}_2(\Omega),W_2)\), where \(W_2\) is the \(2\)-Wasserstein distance, forms a non-complete geodesic metric space. Consider the entropy functional \(\Phi:\mathcal{P}_2(\Omega)\to\mathbb{R}\cup\{+\infty\}\) given by \[\begin{align} \Phi(\nu):= \begin{cases} \displaystyle \int_\Omega \rho(x)\log\rho(x)\,dx & \text{if }\,\, \nu = \rho\,dx \\[1.2ex] +\infty & \text{otherwise.} \end{cases} \end{align}\] It is clear from definition that \(\Phi\) is proper, geodesically convex, and bounded from below. It can further be seen from standard arguments, e.g., [31], that

• \(\Phi\) is lower semicontinuous;

• For any \(\mu\in \mathcal{P}_2(\Omega)\) and \(t>0\), \(\operatorname{prox}_{t\Phi}(\mu)\neq\emptyset.\)

From this example we see that Theorem 1 can replace the usual variational principle on a non-complete geodesic space for certain functionals.

1.4.3 Non-existence of proximal points↩︎

Let \((c_{00},\|\cdot\|_{\ell^2})\) be the space of finitely supported real sequences equipped with the \(\ell^2\)-norm. Consider the functional \(\Phi:c_{00}\to\mathbb{R}\cup\{+\infty\}\) given by \[\Phi(u)=\sum_{n=1}^\infty \frac{n^2}{2}\big(u_n-n^{-2}\big)^2.\] It is clear that \(\Phi\) is proper, convex, and bounded from below. It was proved in [32] that it is nowhere sub-differentiable. Using this result, one can verify that

• \(\Phi\) is lower semicontinuous;

• For every \(u\in c_{00}\) and \(t>0\), \(\operatorname{prox}_{t\Phi}(u) = \emptyset\).

Item \((i)\) follows from [32], while item \((ii)\) follows from the emptiness of the subdifferential at all points and the sum rule for subdifferentials [33]. This example shows that certain pathologies of non-complete spaces cannot be avoided; here Theorem 1 becomes vacuous as there are no proximal points.

2 Proofs of subsections 1.2 and 1.3↩︎

2.1 Proof of Theorem 1↩︎

Let \(\hat{u}\in X\) and \(\lambda>0\) be such that \[\hat{u}\in \mathop{\mathrm{arg\,min}}_{v \in X} \left\{ \Phi(v) + \frac{\varepsilon}{p\lambda^p} d(v,u)^p \right\}.\] Let us begin proving item \((a)\); we proceed by contradiction. Suppose that \(d(u, \hat{u}) > \lambda\). Since \(X\) is a geodesic metric space, there exists a curve \(\gamma \colon [0,1] \to X\) with \(\gamma(0) = u\) and \(\gamma(1) = \hat{u}\) such that \(d(\gamma(s), u) = s\, d(u, \hat{u})\) for all \(s\in[0,1]\). Define \[t := \frac{\lambda}{d(u, \hat{u})}.\] Since we assumed \(d(u,\hat{u})>\lambda\), it must be that \(t\in(0,1)\). The geodesic convexity of \(\Phi\) gives \[\begin{align} \label{convf0} \Phi(\gamma(t)) \leq (1 - t) \Phi(u) + t \Phi(\hat{u}). \end{align}\tag{4}\] By optimality of \(\hat{u}\), we have \[\begin{align} \Phi(\hat{u}) + \frac{\varepsilon}{p\lambda^p} d(\hat{u},u)^p \le \Phi(\gamma(t)) + \frac{\varepsilon}{p\lambda^p} d(\gamma(t),u)^p. \end{align}\] Combining this with (4 ) gives \[\begin{align} \Phi(\hat{u}) + \frac{\varepsilon}{p\lambda^p} d(\hat{u},u)^p \le (1 - t) \Phi(u) + t \Phi(\hat{u}) + \frac{\varepsilon}{p\lambda^p} d(\gamma(t),u)^p. \end{align}\] As \(\gamma\) is a constant-speed geodesic, we have that \(d(\gamma(t),u)=td(\hat{u},u)=\lambda\) , and hence \[\begin{align} \Phi(\hat{u}) + \frac{\varepsilon}{p\lambda^p} d( \hat{u},u)^p &\le (1 - t) \Phi(u) + t \Phi(\hat{u}) + \frac{\varepsilon}{p\lambda^p} t^p d(u, \hat{u})^p \end{align}\] Using that \(\Phi(u) \leq \Phi(\hat{u}) + \varepsilon\) and simplifying, we get \[\begin{align} \Phi(\hat{u}) + \frac{\varepsilon}{p\lambda^p} d( \hat{u},u)^p &\le (1 - t) (\Phi(\hat{u})+\varepsilon) + t \Phi(\hat{u}) + \frac{\varepsilon}{p\lambda^p} t^p d(u, \hat{u})^p\\ &= \Phi(\hat{u})+\varepsilon\Big(1-\frac{\lambda}{d(u,\hat{u})}\Big) + \frac{\varepsilon}{p}. \end{align}\] Rearranging and simplifying, we obtain \[\begin{align} \frac{1}{p\lambda^p} d( \hat{u},u)^p &\le \frac{d(u,\hat{u})-\lambda}{d(u,\hat{u})} + \frac{1}{p} . \end{align}\] Multiplying both sides by \(\lambda^{p}\) and rearranging, \[\begin{align} \label{preconome} \frac{\lambda-d(u,\hat{u})}{d(u,\hat{u})}\lambda^{p}&\le \frac{\lambda^{p}}{p} - \frac{1}{p} d( \hat{u},u)^p \end{align}\tag{5}\] Define \(\omega:[0,+\infty)\to[0,+\infty)\) as \(\omega(s) = s^p/p\). By convexity of \(\omega\), it follows from (5 ) that \[\frac{\lambda-d(u,\hat{u})}{d(u,\hat{u})}\lambda^{p} \le \omega(\lambda)-\omega(d(u,\hat{u}))\le \omega'(\lambda)(\lambda-d(u,\hat{u}))=\lambda^{p-1}(\lambda-d(u,\hat{u})).\] Dividing both sides by \(\lambda^{p-1}(\lambda-d(u,\hat{u}))<0\), we obtain \(\lambda/d(u,\hat{u})\geq 1\). A contradiction. Let us now proceed with item \((b)\). By optimality of \(\hat{u}\), \[\label{optine2} \Phi(\hat{u})+\frac{\varepsilon}{p\lambda^p}d(\hat{u},u)^p\le \Phi(v)+\frac{\varepsilon}{p\lambda^p}d(v,u)^p\quad\forall v\in X.\tag{6}\] In particular, taking \(v=u\) gives \[\Phi(\hat{u}) \le \Phi(\hat{u})+\frac{\varepsilon}{p\lambda^p}d(\hat{u},u)^p\le \Phi(u).\] Finally, we prove item \((c)\). Fix \(v\in X\setminus\{\hat{u}\}\) and let \(\gamma:[0,1]\to X\) be a constant–speed geodesic with \(\gamma(0)=\hat{u}\) and \(\gamma(1) = v\). For each \(t\in[0,1]\), set \(v_t:=\gamma(t)\). Then, \[\begin{align} \label{cti} \Phi(v_t)\le (1-t)\Phi(\hat{u})+t\Phi(v)\quad\text{and}\quad d(v_t,u)\le d(\hat{u},u)+t\,d(v,\hat{u})\quad \forall t\in(0,1]. \end{align}\tag{7}\] Plugging \(v_t\) into (6 ), using (7 ), and simplifying yields \[t\big(\Phi(\hat{u})-\Phi(v)\big)\le \frac{\varepsilon}{\lambda^{p}} \Big(\frac{d(v_{t},u)^{p}}{p}-\frac{d(\hat{u},u)^{p}}{p}\Big).\] As in the proof of item \((a)\), let \(\omega:[0,+\infty)\to[0,+\infty)\) be given by \(\omega(s) = s^p/p\). By convexity of \(\omega\), we see that, for all \(t\in(0,1]\), \[\begin{align} t\big(\Phi(\hat{u})-\Phi(v)\big)&\le \frac{\varepsilon}{\lambda^{p}} \big(\omega(d(v_t,u)) - \omega(d(\hat{u}, u))\big)\\ &\le \frac{\varepsilon}{\lambda^{p}}\omega'(d(v_t,u))(d(v_t,u)-d(\hat{u},u))\\ &= \frac{\varepsilon}{\lambda^{p}} d(v_t,u)^{p-1}(d(v_t,u)-d(\hat{u},u)). \end{align}\] Using (7 ), we get that for all \(t\in(0,1]\), \[\begin{align} t\big(\Phi(\hat{u})-\Phi(v)\big)&\le \frac{\varepsilon}{\lambda^{p}} d(v_t,u)^{p-1} td(v, \hat{u}). \end{align}\] This leads to \(\Phi(\hat{u})-\Phi(v)\le\varepsilon/\lambda^{p}d(v_t,u)^{p-1}d(v, \hat{u})\) for all \(t\in(0,1]\). Letting \(t\longrightarrow0^+\), \[\begin{align} \label{ineqts} \Phi(\hat{u}) - \Phi(v)\le\frac{\varepsilon{\lambda^{p}}}{d}(\hat{u},u)^{p-1}d(v, \hat{u}). \end{align}\tag{8}\] Combining this with the estimate in item \((a)\) yields \[\begin{align} \Phi(\hat{u}) \le \Phi(v) + \frac{\varepsilon}{\lambda^{p}}d(\hat{u},u)^{p-1}d(v,\hat{u}) \le \Phi(v) + \frac{\varepsilon}{\lambda} d(\hat{u}, v). \end{align}\] This finishes the proof, as \(v\in X\setminus\{\hat{u}\}\) was arbitrary. \(\square\)

2.2 Proof of Theorem 7↩︎

From Theorem 1 we obtain items \((a)\) and \((c)\) directly. Proceeding as in the proof of Theorem 1, one arrives at inequality (8 ) for any \(v\in X\setminus\{\hat{u}\}\). From this, the definition of metric slope and (8 ), we obtain \[\begin{align} \label{iNg} |\partial \Phi|(\hat{u}) &\leq \frac{\varepsilon}{\lambda^p} d(\hat{u},u)^{p-1}, \end{align}\tag{9}\] then \(\frac{\lambda^p}{\varepsilon} |\partial \Phi|(\hat{u})\leq d(\hat{u},u)^{p-1}.\) Raising both sides to the power \(q=\frac{p}{p-1}\) yields \[\begin{align} \label{ineq321} \left( \frac{\lambda^p}{\varepsilon} |\partial \Phi|(\hat{u}) \right)^{q} \leq d(\hat{u},u)^p. \end{align}\tag{10}\] Using inequality (10 ) into \[\Phi(\hat{u})+\frac{\varepsilon}{p\lambda^p}d(\hat{u},u)^p\le \Phi(u),\] it follows \[\Phi(\hat{u}) + \frac{\varepsilon}{p\lambda^p} \left( \frac{\lambda^p}{\varepsilon} |\partial \Phi|(\hat{u}) \right)^{q} \leq \Phi(\hat{u}) + \frac{\varepsilon}{p\lambda^p} d(\hat{u},u)^p \leq \Phi(u).\] Thus, we obtain \[\Phi(\hat{u}) + \frac{ \lambda^{q} }{p\varepsilon^{\frac{q}{p}}} |\partial \Phi|(\hat{u})^{q} \leq \Phi(u).\] This completes the proof. \(\square\)

2.3 Proof of Proposition 10↩︎

Let \(u \in \mathop{\mathrm{dom}}\Phi\) and \(t>0\) such that \(\Phi(u) < \inf_{v\in X}\Phi(v) + t^{\frac{q}{p}}\). Let \[\begin{align} \hat{u}\in\mathop{\mathrm{arg\,min}}_{v \in X} \left\{ \Phi(v) + \frac{1}{pt} d(v,u)^p \right\}. \end{align}\] If \(\Phi(u) = \inf_{v\in X}\Phi(v)\), then \(\hat{u}\in \mathop{\mathrm{arg\,min}}_{v\in X}\Phi(v)\), and hence by (?? ), \(d(\hat{u},T(\hat{u}))\le \Phi(\hat{u})-\Phi(T(\hat{u}))\le 0\); so \(\hat{u}\) is a fixed point. Suppose now that \(\Phi(u) - \inf_{v\in X}\Phi(v)>0\). Define \(\varepsilon:=\Phi(u) - \inf_{v\in X}\Phi(v)\). Set \(\lambda = \sqrt[p]{t\varepsilon}\) and observe that \((pt)^{-1} = \varepsilon (p\lambda^p)^{-1}\). We see that the conditions in Theorem 1 are satisfied. By Theorem 1, we have \[\begin{align} \Phi(\hat{u})\le \Phi(v) + \frac{\varepsilon}{\lambda} d(v,\hat{u}) \quad \forall v\in X. \end{align}\] Combining this with (?? ), we conclude that \[\begin{align} d(\hat{u}, T(\hat{u})) + \Phi(T(\hat{u}))\le \Phi(\hat{u})\le \Phi(T(\hat{u})) + \frac{\varepsilon}{\lambda} d(T(\hat{u}),\hat{u}). \end{align}\] This can be rewritten as \[\begin{align} \big(1-\frac{\varepsilon}{\lambda}\big) d(\hat{u}, T(\hat{u})) \le 0. \end{align}\] Since \(\varepsilon/\lambda = \varepsilon^{1-\frac{1}{p}}/t^{\frac{1}{p}} = (\varepsilon^{\frac{p}{q}}/t)^{\frac{1}{p}}<1\), the result follows.

2.4 Proof of Proposition 11↩︎

Set \(\Lambda:=\mathop{\mathrm{arg\,min}}_{v\in X}\Phi(v)\), assume that (?? ) holds and fix \(t>0\). We will prove that for any \(\delta \in (0,1)\) there holds \[\begin{align} \label{etdelta} \Phi(u) -\inf_{v\in X}\Phi(v) \ge \frac{t^{q/p}(1-\delta)^p}{(t^{q/p}+\kappa)^{p}} d\big(u,\Lambda\big)^p \quad \forall u\in\mathop{\mathrm{dom}}\operatorname{prox}_{t\Phi} . \end{align}\tag{11}\] The result will follow by letting \(\delta\longrightarrow0^+\). We proceed by contradiction; suppose there exist \(\delta>0\) and \(u\in\mathop{\mathrm{dom}}\operatorname{prox}_{t\Phi}\) such that (11 ) does not hold. Then, \[\begin{align} \Phi(u) -\inf_{v\in X}\Phi(v) < \frac{t^{q/p}(1-\delta)^p}{(t^{q/p}+\kappa)^{p}} d\big(u,\Lambda\big)^p . \end{align}\] Since \(u\in \mathop{\mathrm{dom}}\operatorname{prox}_{t\Phi}\), there exists \(\hat{u}\in X\) such that \[\hat{u}\in \mathop{\mathrm{arg\,min}}_{v\in X}\big\{ \Phi(v) + \frac{1}{pt} d(v, u)^p\big\}.\] Consider the positive numbers \[\lambda:= \frac{1-\delta}{1+\kappa t^{-q/p}}d(u,\Lambda)\quad \text{and} \quad \varepsilon:=\lambda^{p}/t.\] Observe that by construction, \[\begin{align} \frac{\varepsilon}{p\lambda^p} = \frac{1}{pt}\quad \text{and}\quad \frac{\varepsilon}{\lambda} = \frac{\lambda^{p-1}}{t}. \end{align}\] Since \(\Phi(u)\le\inf_{v\in X}\Phi(v) + \varepsilon\), Theorem 7 yields that \(d(u,\hat{u})\leq\lambda\) and \[\begin{align} |\partial\Phi|(\hat{u}) \le \frac{\lambda^{p-1}}{t}. \end{align}\] This and condition (?? ) yield \[d\big(\hat{u},\Lambda\big) \le \kappa |\partial\Phi|(\hat{u})^{q/p} \le \kappa \Big(\frac{\lambda^{p-1}}{t}\Big)^{q/p} = \lambda\kappa t^{-q/p}.\] Combining the previous inequality with the estimate \(d(u,\hat{u})\le\lambda\), we obtain \[\begin{align} d(u,\Lambda) &\le d(u,\hat{u}) + d(\hat{u},\Lambda)\le \lambda +\lambda\kappa t^{-q/p}\\ &=\lambda(1+\kappa t^{-q/p})=\frac{1-\delta}{1+\kappa t^{-q/p}}d(u,\Lambda)(1+\kappa t^{-q/p})\\ & =(1-\delta)d(u,\Lambda)<d(u,\Lambda). \end{align}\] This is the sought contradiction. \(\square\)

3 The theorem in the space of probability measures↩︎

3.1 Preliminaries and notation↩︎

In this subsection, we recall basic notions from the theory of optimal transport and the structural properties of the Wasserstein space, following [18].

3.1.1 The space of probability measures↩︎

The space of Borel probability measures \(\mu:\mathcal{B}(\mathbb{R}^d)\to [0,1]\) is denoted by \(\mathcal{P}(\mathbb{R}^d)\). For a probability measure \(\mu\in\mathcal{P}(\mathbb{R}^d)\) and a Borel mapping \(\mathfrak t:\mathbb{R}^d\to\mathbb{R}^d\), the pushforward of \(\mu\) through \(\mathfrak t\) is the probability measure \(\mathfrak t_{\#}\mu\in\mathcal{P}(\mathbb{R}^d)\) defined by \[\mathfrak t_\#\mu(B):=\mu(\mathfrak t^{-1}(B)) \quad\forall B\in\mathcal{B}(\mathbb{R}^d).\] A probability measure \(\gamma\in\mathcal{P}(\mathbb{R}^d\times\mathbb{R}^d)\) is said to be a coupling between \(\mu\in\mathcal{P}(\mathbb{R}^d)\) and \(\nu\in\mathcal{P}(\mathbb{R}^d)\) if \[{\operatorname{proj}_1}_{\#}\gamma=\mu\quad \text{and}\quad{\operatorname{proj}_2}_{\#}\gamma=\nu,\] where \(\operatorname{proj}_1,\operatorname{proj}_2:\mathbb{R}^d\times\mathbb{R}^d\to\mathbb{R}^d\) denote the canonical projections. The set of all couplings between \(\mu\) and \(\nu\) is denoted by \(\Gamma(\mu,\nu)\). A sequence \((\mu_n)\subset\mathcal{P}(\mathbb{R}^d)\) is said to converge narrowly to \(\mu\in\mathcal{P}(\mathbb{R}^d)\) if \[\int_{\mathbb{R}^d} \varphi\,\mathrm d\mu_n \longrightarrow \int_{\mathbb{R}^d} \varphi\,\mathrm d\mu \quad \forall \varphi\in C_b(\mathbb{R}^d),\] where \(C_b(\mathbb{R}^d)\) denotes the space of bounded continuous functions \(\varphi:\mathbb{R}^d\to\mathbb{R}\).

3.1.2 The Wasserstein space distance↩︎

Given \(\mu\in\mathcal{P}(\mathbb{R}^d)\), \(L^2(\mu)\) denotes the usual space of square-integrable Borel functions \(f:\mathbb{R}^d\to\mathbb{R}\). We denote \[\mathcal{P}_2(\mathbb{R}^d):= \{\mu\in\mathcal{P}(\mathbb{R}^d):\, \operatorname{id}\in L^2(\mu)^d \}.\] This is precisely the space of Borel probability measures with finite second moment. The Wasserstein distance between \(\mu\in\mathcal{P}_2(\mathbb{R}^d)\) and \(\nu\in\mathcal{P}_2(\mathbb{R}^d)\) is defined by \[W_2(\mu,\nu) :=\inf_{\gamma\in\Gamma(\mu,\nu)} \left(\int_{\mathbb{R}^d\times\mathbb{R}^d}|x-y|^2\,\mathrm d\gamma(x,y)\right)^{\!1/2}.\] The tuple \((\mathcal{P}_2(\mathbb{R}^d),W_2)\) forms a complete and separable metric space.

3.1.3 Geodesics in the Wasserstein space↩︎

Given \(\mu,\nu\in\mathcal{P}_2(\mathbb{R}^d)\), we consider the set \[\Gamma_o(\mu,\nu):=\left\{ \gamma\in\Gamma(\mu,\nu):\,\, W_{2}(\mu,\nu) ^2=\int_{\mathbb{R}^d\times\mathbb{R}^d}|x-y|^2\,\mathrm d\gamma(x,y) \right\}.\] Given \(\mu,\nu\in\mathcal{P}_2(\mathbb{R}^d)\), the set \(\Gamma_o(\mu,\nu)\) is always nonempty. We say that a Borel mapping \(\mathfrak t:\mathbb{R}^d\to\mathbb{R}^d\) is an optimal transport between \(\mu\in\mathcal{P}_2(\mathbb{R}^d)\) and \(\nu\in\mathcal{P}_2(\mathbb{R}^d)\) if \((\operatorname{id},\mathfrak t)_{\#}\mu\) belongs to \(\Gamma_o(\mu,\nu)\). A curve \(\gamma:[0,1]\to\mathcal{P}_2(\mathbb{R}^d)\) is said to be a Wasserstein geodesic if there exists \(\pi\in\Gamma_o(\mu,\nu)\) such that \[\gamma(t) = \big((1-t)\operatorname{proj}_1 + t\operatorname{proj}_2\big)_{\#}\pi \quad \forall t\in[0,1].\] It can be seen that Wasserstein geodesics are precisely the constant-speed geodesics in \((\mathcal{P}_2(\mathbb{R}^d),W_2)\), and that this space is geodesic.

3.1.4 The regular subdifferential↩︎

Let \(\Phi:\mathcal{P}_2(\mathbb{R}^d)\to \mathbb{R}\cup\{+\infty\}\) be a proper geodesically convex functional such that \[\begin{align} \label{absconass} \mathop{\mathrm{dom}}\Phi\subseteq \bigl\{ \mu \in \mathcal{P}_2(\mathbb{R}^d) : \mu = \rho\, dx \text{ for some } \rho \in L^1(\mathbb{R}^d) \bigr\}. \end{align}\tag{12}\] Given \(\mu,\nu\in\mathop{\mathrm{dom}}\Phi\) we will denote by \(\mathfrak t_{\mu}^\nu:\mathbb{R}^d\to\mathbb{R}^d\) the unique optimal transport between \(\mu\) and \(\nu\); the existence and uniqueness is guaranteed by the Brenier-McCann theorem. The subdifferential \(\partial\Phi(\mu)\) of \(\Phi\) at a probability measure \(\mu\in\mathcal{P}_2(\mathbb{R}^d)\) consists of all \(\xi\in L^2(\mu)^d\) such that \[\Phi(\nu) \ge \Phi(\mu) + \int_{\mathbb{R}^d} \langle \xi(x), \mathfrak t_{\mu}^\nu(x)-x\rangle\,d\mu(x)\quad \forall \nu\in\mathcal{P}_2(\mathbb{R}^d).\]

3.2 The Brøndsted-Rockafellar theorem in the Wasserstein space↩︎

Let \(\Phi:\mathcal{P}_2(\mathbb{R}^d)\to \mathbb{R}\cup\{+\infty\}\) be a proper geodesically convex functional bounded from below such that (12 ) holds. Given \(\tau>0\), we consider the set-valued mapping \(J_{\tau}^\Phi:\mathcal{P}_2(\mathbb{R}^d)\rightrightarrows\mathcal{P}_2(\mathbb{R}^d)\) given by \[J_{\tau}^\Phi(\mu) :=\mathop{\mathrm{arg\,min}}_{\nu\in\mathcal{P}_{2}(\mathbb{R}^d)}\left\{ \Phi(\nu) + \frac{1}{2\tau} W_2(\nu,\mu)^2 \right\}.\] This is precisely the proximal mapping (for \(p=2\)) introduced in Subsection 1.2; it also corresponds to a single implicit–Euler step of the minimizing movement scheme for gradient flows, often referred to as the Jordan–Kinderlehrer–Otto (JKO) scheme (see [34] and [18]). We now formulate the metric analogue of the Brøndsted–Rockafellar theorem in the Wasserstein framework, and give a couple of remarks before the proof.

3.3 Entropy-transportation and metric sub-regularity↩︎

Let \(\Phi:\mathcal{P}_2(\mathbb{R}^d)\to \mathbb{R}\cup\{+\infty\}\) be a proper geodesically convex functional such that (12 ) holds. Given \(\mu\in\mathcal{P}_2(\mathbb{R}^d)\) and \(C\subseteq\mathcal{P}_2(\mathbb{R}^d)\), we denote \[W_2\big(\mu,C\big):=\inf_{\nu\in C}W_2(\mu,\nu)\quad\text{and}\quad d_{L^2}\big(0,\partial\Phi(\mu)\big):=\inf_{\xi\in\partial\Phi(\mu)}\|\xi\|_{L^2(\mu)^d}.\] The first quantity can be interpreted as a gauge of how close a given probability measure is to belonging to a specified set, while the second—being the minimal \(L^2\)-norm among all elements of the subdifferential of \(\Phi\) at \(\mu\)—serves as a gauge of how close \(\mu\) is to having zero as a subgradient, and hence to being a minimizer of \(\Phi\). As a consequence of Theorem 12, we obtain an equivalence between the metric sub-regularity of the subdifferential and a quantitative growth condition of entropy–transport type.

4 The particular case of normed spaces↩︎

In this subsection of the appendix, we specialize the metric constructions of Section 1.2 to the classical setting of a real normed space. In this framework, the optimality condition for a proximal step admits a concrete expression in dual variables, and it becomes transparent how the choice of a proximal point produces an approximate minimizer together with a subgradient of controlled size. The following theorem can be regarded as an extension of [4] from Hilbert spaces to general normed spaces. Its proof is a direct consequence of Theorem 1.

5 The limiting case \(p=+\infty\)↩︎

This part of the appendix records the limiting regime in which the proximal regularization becomes a hard constraint. As the exponent in the powered-distance kernel tends to infinity, the penalized problems enforce confinement to a fixed ball; in the sense of \(\Gamma\)–convergence the kernels converge to the indicator function of a ball. Accordingly, the approximate minimizer produced by the proximal step is replaced by a minimizer under a ball constraint, and the same Ekeland-type estimate is recovered.

6 The case of general proximal kernels↩︎

In this subsection of the appendix, we present a more general version of Theorem 1, in which powered-distance perturbations are replaced by more general ones. Let \(\omega:[0,+\infty)\to[0,+\infty)\) be a convex function satisfying \[\begin{align} \omega(0)=0\qquad\text{and}\qquad \omega(s)>0\quad \forall s>0. \end{align}\] These properties imply the existence of the right-derivative \(\omega'_+:[0,+\infty)\to[0,+\infty)\) and furthermore that \(\omega_+'(s)>0\) for all \(s>0\). We are now ready to state the general theorem.

References↩︎