Approximation closure for the moment-Hessian constant

Part of the moment-map mechanism, Chapter The moment map: the deterministic inequality; the reading order is on the full proofs page.

Overview. This dossier answers Proposition 15.1 (Theorem D27.1). Every centered log-concave law, including one carried by a proper affine subspace, is a $\Wtwo$-limit of explicit regular compact-target approximants $\mu_k$ with $\CPaff(\mu)\le\liminf_k\CPaff(\mu_k)\le\liminf_k\CMH(\mu_k)$ ((D27.5)). A uniform CMH bound on these approximants ((D27.6)) therefore gives the affine Poincaré inequality for $\mu$ with the same constant. That uniform bound is an explicit hypothesis: no universal $\mathrm{CMH}(4)$, no KLS and no continuity of $\CMH$ is claimed. Only the Poincaré inequality is passed to the limit, never the Stein kernels.

1. Gaussian smoothing, a quadratic tilt, restriction to a ball and centering give $\mu_k$ with $\Wtwo(\mu_k,\mu)<2/k+\sqrt n/k^2$ ((D27.15)). Their potentials are smooth and strictly convex ((D27.18)).

2. Theorem 4.1 and Fathi’s Stein theorem supply the kernel $H_k$ with $\E_{\mu_k}H_k=\Sigma_k$ ((D27.22)). The Stein form on the ambient core is closable with constant kernel (Lemma D27.1), so Theorem 16.1 gives (D27.28).

3. At a singular limit, the intrinsic covariance form on $S=\operatorname{Ran}\Sigma$ is closable and $\R+C_c^\infty(S)$ is a dense core. It agrees with the ambient core because $\Sigma$ and $\Sigma^+$ annihilate normal derivatives ((D27.32)).

4. Convergence of the covariances, variances and energies on the common core ((D27.33)–(D27.35)) with the density of step 3 gives the lower semicontinuity of $\CPaff$. Combined with step 2 this gives (D27.5).

5. Any whitening is applied only to individual approximants and undone before the limit, since whitening diverges in collapsing directions ((D27.38)).

Scope. This dossier answers Proposition 15.1 with the closed-form convention already used by Definition 16.1 and Theorem 16.1. It constructs regular compact-target approximants of every centered log-concave law, including a law carried by a proper affine subspace, and proves that a uniform CMH estimate on those approximants passes to the affine Poincaré inequality with no loss. The uniform CMH estimate is an explicit hypothesis: nothing below proves universal $\mathrm{CMH}(4)$ or Conjecture 0.1, and no continuity of $\CMH$ is asserted.

1. Statement and the limiting closed form¶

Let $\mu$ be a centered log-concave probability measure on $\R^n$, let $\Sigma=\Cov(\mu)$, and put

The affine hull of $\mu$ is $S$. Indeed, if $v\in\ker\Sigma$, then $\Var_\mu\inner{v}{X}=0$ and centeredness gives $\inner{v}{X}=0$ almost surely, so the affine hull is contained in $(\ker\Sigma)^\perp=\operatorname{Ran}\Sigma$. Conversely, the covariance is positive definite on the affine hull: otherwise the support would lie in a proper affine hyperplane of its own affine hull. Thus $\Sigma_S\succ0$ when $d>0$.

For $d>0$, start on the globally Lipschitz functions $f:S\to\R$ that belong to $L^2(\mu)$ with the covariance pre-form

Its closed relaxation in $L^2(\mu)$ is denoted by $\calE_{\Sigma,\mu}$, and its domain by $H^1_\Sigma(\mu)$. This is the intrinsic convention throughout the dossier; no equality with a separately defined maximal distributional or Neumann Sobolev domain is claimed. Section 4 below proves both closability and density of

in this domain. Define

If $d=0$, then $\mu=\delta_0$; by convention $H^1_\Sigma(\mu)=L^2(\mu)=\R$ and $\CPaff(\mu)=0$.

2. Construction of regular compact-target approximants¶

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

The density $q_\delta$ is positive and $C^\infty$ on $\R^n$. It is log-concave by the Prékopa theorem (equivalently, convolution preserves log-concavity). Moreover

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

Let $m_{\delta,\eps,R}$ be its mean and center it:

For fixed $\delta$, as $\eps\downarrow0$ and $R\uparrow\infty$, the density ratio in (D27.10) converges pointwise to one relative to $\lambda_\delta$ and is bounded above by $1/Z_{\delta,\eps,R}$, with $Z_{\delta,\eps,R}\to1$. Dominated convergence therefore gives weak convergence to $\lambda_\delta$ and convergence of the second moment: for the latter, apply it to $|x|^2\one_{B(0,R)}e^{-\eps|x|^2/2}$, dominated by the $\lambda_\delta$-integrable function $|x|^2$. The weak-plus-second-moment characterization of quadratic Wasserstein convergence gives

Here is an explicit diagonal choice. Set $\delta_k=k^{-4}$. By (D27.12), choose

and put $m_k=m_{\delta_k,\eps_k,R_k}$ and $\mu_k=\mu_{\delta_k,\eps_k,R_k}$. Since $\lambda_{\delta_k}$ is centered, comparison of means under any coupling gives

Translation by $-m_k$, the triangle inequality, and (D27.9) now yield the quantitative diagonal estimate

No whitening has been used.

We next check every target hypothesis. The support of $\mu_k$ is the convex body

and, on its interior,

The function $g_k$ extends to a positive $C^\infty$ function on all of $\R^n$. The measure is centered by construction and full-dimensional because $g_k>0$ on the nonempty open ball $\operatorname{int}P_k$. In particular, its covariance $\Sigma_k$ is positive definite. Also $0\in\operatorname{int}P_k$: the barycenter of a positive density on a full-dimensional convex body cannot lie on a supporting hyperplane of that body.

Writing $V_k=-\log g_k$ on $\operatorname{int}P_k$, log-concavity of $q_{\delta_k}$ gives

Thus $\mu_k$ is log-concave, and its interior potential is smooth and strictly convex. (The additional standard convolution identity gives $D^2(-\log q_{\delta_k})\preceq\delta_k^{-1}I_n$, but no uniform Hessian bound is used.)

The published compact-target regularity theorem Berman & Berndtsson, 2013, in the repository form of Theorem 4.1, now applies to the centered probability $g_k\one_{\operatorname{int}P_k}\,\dd x$. It supplies a smooth strictly convex canonical moment potential $\varphi_k$ on $\R^n$ and a global diffeomorphism

With target coordinate $x=\nabla\varphi_k(y)$, define

Fathi’s moment-map Stein theorem Fathi, 2019 gives a smooth positive symmetric field on $\operatorname{int}P_k$ satisfying, for every scalar ambient test $F\in C_c^\infty(\R^n)$,

Equivalently, $\operatorname{Div}_{\mu_k}H_k=-x$ distributionally on the ambient space. The absence of a boundary distribution in (D27.21) is precisely the weak zero-normal-flux convention used by the certified endpoint. Since $P_k$ is bounded, testing with a cutoff that equals the coordinate function $x_j$ near $P_k$ gives

Thus every hypothesis of the imported regular moment-map input has been matched.

3. The exact closed Stein form on each approximant¶

For fixed $k$, let

and define on this ambient restriction core

The value is independent of the ambient extension, because two smooth extensions agreeing on the open set $\operatorname{int}P_k$ have the same gradient there. The core is dense in $L^2(\mu_k)$, and (D27.22) shows that every core function has finite energy:

The weak Stein identity gives, on smooth tests for which the displayed quantities are integrable,

Thus the Friedrichs form operator in Lemma D27.1 is exactly the closed-form object used by Definition 16.1; its inverse is understood only on the orthogonal complement of its constant kernel. We do not identify this domain with an unnamed maximal Neumann or maximal distributional domain.

Theorem 16.1 is therefore applicable to every $\mu_k$ and gives

This is the only step at which the CMH quantity is used.

4. The intrinsic domain at a singular limit¶

A log-concave probability whose affine hull is $S$ has a density with respect to $d$-dimensional Lebesgue measure on $S$; that density is positive and locally bounded above and below on the relative interior of its convex support. Consequently the same local distributional-derivative argument as in Lemma D27.1, now with the constant positive matrix $\Sigma_S$, proves that the pre-form (D27.2) is closable. It also shows that the kernel of its closure consists of the constants, because the relative interior of a convex support is connected.

We now prove the promised core density rather than assume it. It is enough to approximate a globally Lipschitz $f\in L^2(\mu)$ in the form norm.

(1) Let $T_Mf=(-M)\vee f\wedge M$. The Lipschitz chain rule and dominated convergence give $T_Mf\to f$ in $L^2(\mu)$ and $\calE_{\Sigma,\mu}^0(T_Mf-f)\to0$.

(2) For bounded $f$, choose $\chi_R\in C_c^\infty(S)$ equal to one on $B_S(0,R)$, supported in $B_S(0,2R)$, with $|\nabla_S\chi_R|\le c/R$. Then $\chi_Rf\to f$ in $L^2(\mu)$ and in energy. Indeed, the tail part containing $(1-\chi_R)\nabla_Sf$ tends to zero by dominated convergence, while

(3) A compactly supported Lipschitz function on the Euclidean space $S$ can be mollified inside $S$. The mollifications converge uniformly in value, their gradients converge Lebesgue-almost everywhere, and their gradients are uniformly bounded by the original Lipschitz constant. Since $\mu$ is absolutely continuous on $S$, dominated convergence gives convergence in both $L^2(\mu)$ and the covariance energy.

The resulting functions belong to $C_c^\infty(S)$, proving that $\mathscr C_S$ is dense in $H^1_\Sigma(\mu)$. When $d=0$, the intrinsic space is the singleton space and this statement reduces to density of the constants.

This intrinsic core is exactly the restriction of the one common ambient core

Indeed, an ambient test restricts to an element of $\mathscr C_S$. Conversely, in the orthogonal splitting $x=s+t\in S\oplus S^\perp$, extend $\phi\in C_c^\infty(S)$ by

where $\eta\in C_c^\infty(S^\perp)$ equals one near 0. Then $F\in C_c^\infty(\R^n)$, $F|_S=\phi$, and its normal derivative vanishes on $S$.

More generally, if two ambient extensions agree on $S$, their tangential gradients agree and the difference of their ambient gradients lies in $S^\perp$. With $P_S$ denoting orthogonal projection,

Hence ambient restriction and intrinsic evaluation agree in the covariance energy, and both $\Sigma$ and its Moore–Penrose inverse annihilate every normal derivative. This is exactly the affine-tangent convention in Definition 16.1; no inverse is taken in a collapsing normal direction.

5. Constant-preserving lower semicontinuity¶

Quadratic Wasserstein convergence in (D27.15) implies weak convergence together with convergence of second moments. Since all measures are centered,

in every matrix norm (apply second-moment uniform integrability to each entry, or use polarization).

Fix one $F\in\mathscr C$. Both $F$ and $F^2$ are bounded and continuous, so

For the energies, set $h(x)=\inner{\Sigma\nabla F(x)}{\nabla F(x)}$, a bounded continuous function. Then

Let $L=\liminf_k\CPaff(\mu_k)$ and select a subsequence along which the constants converge to $L$. If $L=\infty$ there is nothing to prove. Otherwise, the affine Poincaré inequality on each approximant, followed by (D27.34)– (D27.35), gives on the common core

Restricting to $S$ and using the core density proved in Section 4 extends the inequality to every $f\in H^1_\Sigma(\mu)$. Therefore

Combining this with the pointwise certified endpoint (D27.28) proves (D27.5). In particular, (D27.6) yields $\CPaff(\mu)\le C$ with no change of constant. Notice that the argument passes only the Poincaré/Dirichlet-form inequality. It uses neither convergence of $H_k$ nor lower semicontinuity of $\CMH$.

6. Eigenvalues, whitening, and affine-support collapse¶

Order covariance eigenvalues decreasingly. If $\lambda_1(\Sigma)\ge\cdots\ge\lambda_d(\Sigma)>0$ are the positive eigenvalues, followed by $n-d$ zeros, then (D27.33) and Weyl’s inequality give

Every $\Sigma_k$ is positive definite, so each approximant may be whitened by the invertible map $A_k=\Sigma_k^{-1/2}$. Invertible affine covariance of both $\CMH$ and $\CPaff$ allows one to apply the regular endpoint in those isotropic coordinates and then pull its inequality back to $\mu_k$.

When $d<n$, however, (D27.38) shows that $A_k$ diverges in the collapsing normal directions. The whitened laws all have covariance $I_n$ and therefore cannot converge in $\Wtwo$ to a law with covariance $\Sigma$ singular. The valid order is:

(1) whiten an individual full-dimensional approximant if desired;

(2) use invertible affine covariance to unwhiten its Poincaré inequality;

(3) take the $\Wtwo$ limit in the original coordinates using (D27.35).

No canonical moment-map kernel is pushed through a noninvertible limiting map. At the limit, only the intrinsic covariance form on $S=\operatorname{Ran}\Sigma$ remains, with normal directions removed by (D27.32).

Obstructions respected. The ledger gives prop:cmh-approximation-closure no bounded_by edge. The argument also does not enter the fixed-cut obstruction regime: it uses no tail split, projection-only test, localization occupation estimate, relative trace upgrade, evolving isoperimetric competitor, or rank-one product-cut assertion. In particular, affine-support collapse is handled at the Poincaré form level rather than by transporting a canonical Stein kernel through a noninvertible map.

Conditional status and exclusions. The approximation sequence exists unconditionally, and the closure theorem is analytic. The only unresolved premise in the route implication is (D27.6). Thus this dossier can at most certify the conditional node “uniform CMH on the constructed regular approximants implies the affine Poincaré inequality for the limit.” CMH is used only as a sufficient condition. No converse, no universal $\mathrm{CMH}(4)$ estimate, and no proof of KLS appears here.