Lower semicontinuity of the affine Poincaré constant

Part of the moment-map mechanism, Chapter The moment map: CMH and the linear test; the reading order is on the full proofs page.

Overview. This dossier proves Lemma 16.3 and, conditional on the unresolved Assumption 16.1, Corollary 16.3. It works with the intrinsic covariance form on the affine support, which makes the affine Poincaré constant $\CPaff$ lower semicontinuous along every centered log-concave $W_2$ -convergent sequence. Regular compact-target approximants come from smoothing, tilting and truncation, together with the published moment-map theorem Theorem 4.1. No bound on $\CMH$ and no semicontinuity of $\CMH$ is claimed.

Lemma D9.1: the covariance pre-form (D9.2) is closable and its kernel is the constants. The class $\R+C_c^\infty$ is a common intrinsic and ambient core, and (D9.8) makes the energy independent of the extension.
Theorem D9.1 (i): Gaussian smoothing, Gaussian tilt, ball truncation and recentering (D9.12) give regular compact-target laws with (D9.16). Theorem 4.1 then supplies the kernel (D9.19).
Theorem D9.1 (ii): convergence of covariances, variances and energies on the core, (D9.21) and (D9.22), gives (D9.9) by Step 1, including for singular $\Sigma$ .
For regular laws, the closed Stein form matches Definition 16.1, so Theorem 16.1 gives (D9.25).
Corollary D9.1: Steps 3–4 applied to the sequence supplied by Assumption 16.1 give $\CPaff\le C$ , which is Conjecture 0.1. This step is conditional on that assumption.

Scope and certification boundary. The first result below proves Lemma 16.3. In fact, its lower-semicontinuity assertion is proved along every centered log-concave $W_2$ -convergent sequence, while the existence of compact-target regular approximants uses the published moment-map result recorded as Theorem 4.1. The second result proves Corollary 16.3 under the unresolved Assumption 16.1. No bound on $\CMH$ is proved, and no convergence or semicontinuity of $\CMH$ is asserted.

1. The covariance form on an affine support¶

Let $\mu$ be a centered log-concave probability on $\R^n$ , set

\Sigma=\Cov(\mu),\qquad S=\operatorname{Ran}\Sigma,\qquad d=\dim S,

(D9.1)

and, when $d>0$ , write $\Sigma_S=\Sigma|_S$ . The affine hull of $\mu$ is the linear space $S$ . Indeed, if $v\in\ker\Sigma$ , then centeredness and $\Var_\mu\langle v,X\rangle=0$ give $\langle v,X\rangle=0$ almost surely. Conversely, the covariance is positive definite on the linear span of the centered support. Thus $\Sigma_S\succ0$ for $d>0$ .

On the globally Lipschitz functions on $S$ that belong to $L^2(\mu)$ , consider the pre-form

\calE^0_{\Sigma,\mu}(f) =\int_S\inner{\Sigma_S\nabla_S f}{\nabla_S f}\,\dd\mu.

(D9.2)

We denote its closure by $\calE_{\Sigma,\mu}$ , its closed domain by $H^1_\Sigma(\mu)$ , and define

\CPaff(\mu) =\sup_{f\in H^1_\Sigma(\mu)\setminus\R} \frac{\Var_\mu f}{\calE_{\Sigma,\mu}(f)}.

(D9.3)

When $d=0$ , centeredness gives $\mu=\delta_0$ ; we set $H^1_\Sigma(\mu)=L^2(\mu)=\R$ and $\CPaff(\mu)=0$ .

Proof D9.1

A log-concave probability with affine hull $S$ has a density with respect to Lebesgue measure on $S$ which is positive and locally bounded above and below on the relative interior of its convex support. Suppose that $f_j$ is Lipschitz, $f_j\to0$ in $L^2(\mu)$ , and $\Sigma_S^{1/2}\nabla_Sf_j\to u$ in $L^2(\mu;S)$ . On every compact subset of the relative interior, the density is equivalent to Lebesgue measure and $\Sigma_S^{-1/2}$ is bounded. Hence $f_j\to0$ and $\nabla_Sf_j\to\Sigma_S^{-1/2}u$ in local Lebesgue $L^2$ . Closedness of distributional differentiation gives $u=0$ . This proves closability. The same local argument applied to a zero-energy element of the closure shows that its distributional gradient vanishes on a connected open convex set, so the element is constant almost everywhere.

It remains to verify the asserted core rather than assume it. Start with a Lipschitz $f\in L^2(\mu)$ . The value truncations $T_Mf=(-M)\vee f\wedge M$ converge to $f$ in $L^2$ and in the energy (D9.2), by the Lipschitz chain rule and dominated convergence. For bounded $f$ , choose $\chi_R\in C_c^\infty(S)$ which equals one on $B_S(0,R)$ , is supported in $B_S(0,2R)$ , and satisfies $|\nabla_S\chi_R|\le c/R$ . Then $\chi_Rf\to f$ in $L^2$ , while

\begin{aligned} \calE^0_{\Sigma,\mu}(\chi_Rf-f) &\le 2\int_{S\setminus B_S(0,R)} \inner{\Sigma_S\nabla_Sf}{\nabla_Sf}\,\dd\mu \\ &\quad+\frac{2c^2}{R^2}\norm{f}_\infty^2\norm{\Sigma_S}_\op \longrightarrow0. \end{aligned}

(D9.6)

Finally, a compactly supported Lipschitz function on the Euclidean space $S$ can be mollified inside $S$ . Its mollifications converge in value and gradient almost everywhere, and their gradients are bounded by the original Lipschitz constant. Absolute continuity of $\mu$ on $S$ and dominated convergence give convergence in the form norm. Therefore $\mathscr C_S$ is a core.

An ambient function in $\mathscr C$ plainly restricts to $\mathscr C_S$ . Conversely, in the orthogonal splitting $x=s+t\in S\oplus S^\perp$ , extend $\phi\in C_c^\infty(S)$ by

F(s+t)=\phi(s)\eta(t),

(D9.7)

where $\eta\in C_c^\infty(S^\perp)$ equals one near zero. This is an ambient compactly supported smooth extension. If two ambient extensions agree on $S$ , the difference of their gradients on $S$ lies in $S^\perp$ . Since

\Sigma=P_S\Sigma_SP_S, \qquad \Sigma^+=P_S\Sigma_S^{-1}P_S,

(D9.8)

both $\Sigma$ and its Moore–Penrose inverse annihilate that normal difference. Thus the ambient covariance energy is exactly the intrinsic energy and is independent of the extension.

2. A regular compact-target recovery sequence¶

Theorem D9.1 (Affine-Poincaré

W_2

lower semicontinuity and regular recovery)

Let $\mu$ be a centered log-concave probability on $\R^n$ .

(i) There are centered, full-dimensional, compactly supported log-concave probabilities $\mu_k$ in the regular compact-target moment-map class such that $\Wtwo(\mu_k,\mu)\to0$ in the original ambient coordinates.

(ii) More generally, for every sequence of centered log-concave probabilities $\nu_k$ with $\Wtwo(\nu_k,\mu)\to0$ ,

\CPaff(\mu)\le\liminf_{k\to\infty}\CPaff(\nu_k).

(D9.9)

In particular, the sequence in (i) has the lower-semicontinuity property required by Lemma 16.3, including when $\mu$ is carried by a proper affine subspace. The assertion concerns only covariance Dirichlet forms and makes no claim about $\CMH(\mu_k)$ or its limit.

Proof D9.2

Let $X\sim\mu$ and let $G\sim N(0,I_n)$ be independent. For $\delta>0$ , put

\lambda_\delta=\mathcal L(X+\sqrt\delta G), \qquad q_\delta=\frac{\dd\lambda_\delta}{\dd x}.

(D9.10)

The density $q_\delta$ is positive and smooth, and it is log-concave because convolution preserves log-concavity Brascamp & Lieb, 1976. The displayed coupling also gives

\Wtwo^2(\lambda_\delta,\mu)\le n\delta, \qquad \int x\,\dd\lambda_\delta(x)=0, \qquad \Cov(\lambda_\delta)=\Sigma+\delta I_n.

(D9.11)

For $\delta,\eps>0$ and $R<\infty$ , define

Z_{\delta,\eps,R} =\int_{B(0,R)}q_\delta(x)e^{-\eps|x|^2/2}\,\dd x, \qquad \dd\widetilde\mu_{\delta,\eps,R}(x) =Z_{\delta,\eps,R}^{-1}\one_{B(0,R)}(x) q_\delta(x)e^{-\eps|x|^2/2}\,\dd x,

(D9.12)

and let $m_{\delta,\eps,R}$ be its mean. For fixed $\delta$ , as $\eps\downarrow0$ and $R\uparrow\infty$ , the density ratio in (D9.12) relative to $\lambda_\delta$ tends pointwise to one, its normalizer tends to one, and it is eventually bounded by two. Dominated convergence, first with bounded continuous tests and then with $|x|^2$ , gives weak convergence and convergence of second moments. Hence

\Wtwo(\widetilde\mu_{\delta,\eps,R},\lambda_\delta)\longrightarrow0.

(D9.13)

Take $\delta_k=k^{-4}$ and use (D9.13) to choose $0<\eps_k<k^{-1}$ and $R_k>k$ such that

\Wtwo(\widetilde\mu_{\delta_k,\eps_k,R_k},\lambda_{\delta_k})<k^{-1}.

(D9.14)

Set $m_k=m_{\delta_k,\eps_k,R_k}$ and recenter:

\mu_k=(x\mapsto x-m_k)_\#\widetilde\mu_{\delta_k,\eps_k,R_k}.

(D9.15)

Comparison of means under any quadratic-cost coupling and centeredness of $\lambda_{\delta_k}$ give $|m_k|<k^{-1}$ . Therefore the triangle inequality and (D9.11) yield the ambient estimate

\Wtwo(\mu_k,\mu)<\frac2k+\frac{\sqrt n}{k^2}.

(D9.16)

We verify the regular class before using the words “regular recovery.” The support of $\mu_k$ is the convex body $P_k=B(-m_k,R_k)$ , and on its interior

\dd\mu_k(z)=g_k(z)\,\dd z, \qquad g_k(z)=Z_{\delta_k,\eps_k,R_k}^{-1} q_{\delta_k}(z+m_k)e^{-\eps_k|z+m_k|^2/2}.

(D9.17)

The function $g_k$ extends to a positive smooth function on $\R^n$ . It is log-concave, $\mu_k$ is centered by construction, and it is full-dimensional because $g_k>0$ on the open ball. Its barycenter zero lies in $\operatorname{int}P_k$ : a positive density on a full-dimensional convex body cannot have its barycenter on a supporting hyperplane. The published compact-target theorem Berman & Berndtsson, 2013Fathi, 2019, in the repository form of Theorem 4.1, now supplies a smooth strictly convex moment potential $\varphi_k$ , a diffeomorphism $\nabla\varphi_k:\R^n\to\operatorname{int}P_k$ , and the smooth positive symmetric kernel

H_k(x)=D^2\varphi_k((\nabla\varphi_k)^{-1}(x))

(D9.18)

with weak zero flux,

\Div_{\mu_k}H_k=-x, \qquad \E_{\mu_k}H_k=\Sigma_k:=\Cov(\mu_k).

(D9.19)

Thus (i) is proved.

We next prove the stronger assertion (ii). Write $\Sigma_k^\nu=\Cov(\nu_k)$ . Quadratic Wasserstein convergence gives convergence of second moments. Since the measures are centered, it follows that

\Sigma_k^\nu\longrightarrow\Sigma

(D9.20)

in every matrix norm. For completeness, under couplings with $\E|X_k-X|^2\to0$ , Cauchy–Schwarz gives convergence of each $\E[X_{k,i}X_{k,j}]$ to $\E[X_iX_j]$ , because the second moments of $X_k$ remain bounded.

If $d=0$ , then $\CPaff(\mu)=0$ by convention and (D9.9) is immediate. We may therefore assume $d>0$ for the remaining common-core argument.

Fix $F\in\mathscr C$ from (D9.5). Both $F$ and $F^2$ are bounded and continuous, so weak convergence gives

\Var_{\nu_k}F\longrightarrow\Var_\mu F.

(D9.21)

Also, with $h(x)=\inner{\Sigma\nabla F(x)}{\nabla F(x)}$ , which is bounded and continuous,

\begin{aligned} &\left|\int\inner{\Sigma_k^\nu\nabla F}{\nabla F}\,\dd\nu_k -\int\inner{\Sigma\nabla F}{\nabla F}\,\dd\mu\right| \\ &\quad\le \norm{\Sigma_k^\nu-\Sigma}_\op\norm{\nabla F}_\infty^2 +\left|\int h\,\dd\nu_k-\int h\,\dd\mu\right| \longrightarrow0. \end{aligned}

(D9.22)

Let $L=\liminf_k\CPaff(\nu_k)$ . If $L=\infty$ , there is nothing to prove. Otherwise choose one subsequence, independent of the test function, along which $\CPaff(\nu_k)\to L$ . The affine Poincaré inequality for $\nu_k$ , followed along this fixed subsequence by (D9.21) and (D9.22), gives for every $F\in\mathscr C$

\Var_\mu F \le L\int\inner{\Sigma\nabla F}{\nabla F}\,\dd\mu.

(D9.23)

Lemma D9.1 identifies the restriction of $\mathscr C$ with the intrinsic core on $S$ and extends the inequality in the closed form norm to all $H^1_\Sigma(\mu)$ . Hence $\CPaff(\mu)\le L$ , proving (ii). If $\Sigma$ is singular, (D9.8) removes all normal derivatives; no inverse is taken in a collapsing direction and no whitening is performed before the limit.

3. Verification of the regular CMH endpoint¶

Although Theorem D9.1 itself does not use $\CMH$ , the conditional corollary needs the exact regular endpoint. We record why the compact-target objects above have the closed-form realization required by Definition 16.1.

Here a regular recovery sequence means a sequence of centered, full-dimensional, compact-target moment-map laws carrying the canonical kernel and weak zero-flux convention of Theorem 4.1. For one such law $\nu$ , with target $P$ , density $g$ , covariance $\Sigma_\nu\succ0$ , and canonical kernel $H$ , let

\mathscr D=\{F|_P:F\in\R+C_c^\infty(\R^n)\}, \qquad \calE_H^0(f)=\int_P\inner{H\nabla f}{\nabla f}\,\dd\nu.

(D9.24)

This core is dense in $L^2(\nu)$ because it contains the restrictions of $C_c^\infty(\operatorname{int}P)$ and the convex boundary is null. The form is finite there because $\E_\nu H=\Sigma_\nu$ . It is closable: if $f_j\to0$ in $L^2(\nu)$ and $H^{1/2}\nabla f_j\to u$ in $L^2(\nu;\R^n)$ , then on each compact subset of $\operatorname{int}P$ , the density is bounded above and below and $H^{\pm1/2}$ are bounded. Thus $f_j\to0$ and $\nabla f_j\to H^{-1/2}u$ in local Lebesgue $L^2$ ; closedness of distributional differentiation forces $u=0$ . The same argument shows that a zero-energy element of the closure is constant, because $\operatorname{int}P$ is connected. Finally, the global weak Stein identity in (D9.19) identifies the Friedrichs form operator with the closed Stein generator used in Definition 16.1; there is no hidden boundary distribution or unnamed maximal-domain convention. Consequently the already-certified Theorem 16.1 applies and gives

\CPaff(\nu)\le\CMH(\nu)

(D9.25)

for every member of a regular recovery sequence.

Hypotheses and conditional status. The lemma uses centeredness, log-concavity, finite-dimensionality, and the published regular compact-target moment-map theorem. Centeredness identifies the affine hull with $\operatorname{Ran}\Sigma$ and is preserved by the construction. Log-concavity supplies the intrinsic density and is preserved by smoothing, tilt, convex truncation, translation, and $W_2$ limits. No isotropy, spectral gap, smoothness of the limiting law, or full-dimensionality of the limit is assumed. The corollary additionally uses Definition 16.1, the certified endpoint Theorem 16.1, and the unresolved Assumption 16.1. The lemma is unconditional relative to its published imported moment-map input; the corollary remains conditional precisely on that open recovery-envelope assumption.

Obstructions respected. Neither ledger node has a bounded_by edge. The proof uses no localization occupation estimate, fixed cut, projection-only test, trace upgrade, or evolving isoperimetric competitor. Affine-support collapse is handled solely by ambient $W_2$ convergence and the intrinsic closed covariance form. No canonical moment-map kernel is transported through a noninvertible map, and no lower semicontinuity of $\CMH$ is claimed.

Unclosed step and deferred ledger artifact. The analytic proof of the lemma has no unclosed step. The only unclosed mathematical premise in the corollary is Assumption 16.1. Since this dossier has checked_by: none, it has no ledger value. The future shared solution: solutions/lem-affine-poincare-w2-liminf.md is only a deferred artifact candidate pending independent review; no ledger delta is applicable now.

References¶

Brascamp, H. J., & Lieb, E. H. (1976). On Extensions of the Brunn–Minkowski and Prékopa–Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion Equation. Journal of Functional Analysis, 22(4), 366–389. 10.1016/0022-1236(76)90004-5
Berman, R. J., & Berndtsson, B. (2013). Real Monge–Ampère Equations and Kähler–Ricci Solitons on Toric Log Fano Varieties. Annales de La Faculté Des Sciences de Toulouse. Mathématiques, 22(4), 649–711. 10.5802/afst.1386
Fathi, M. (2019). Stein Kernels and Moment Maps. The Annals of Probability, 47(4), 2172–2185. 10.1214/18-AOP1305