The exponential cones - The KLS theorem and its methods

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

Overview. This dossier proves Proposition 17.1, Proposition 17.2 and Corollary 17.4 for the exponential cone measures of Definition 17.1. It lifts the moment potential of the uniform law on the base $K$ to an explicit moment potential of the centered cone measure, reads off the canonical Stein kernel, and then computes the linear sector and the cube-cone gate matrix exactly. The outside inputs are Theorem 4.1 for the base, the uniqueness-up-to-translation theorem of Cordero-Erausquin–Klartag (Theorem D21.1), and Gamma moments; Proposition 16.2 and Lemma 16.2 are not used.

Structure of $\mu_{K,\beta}$ as the law of $S(1,U)$ with $S$ Gamma and $U$ uniform on $K$ , with its covariance (Lemma D21.2); base potential and its rescaling (Lemma D21.3).
The cone lift $\varphi$ of (D21.11) is a moment potential of $\bar\mu$ , giving the kernel (D21.13) with $\tau e_1=x$ and $\E\tau=\Sigma$ (Theorem D21.2).
Stein identity on polynomial-growth functions and the radial identity (D21.17) (Lemma D21.4), then domain membership of $\psi$ and $x_1$ in the no-flux closed forms (Lemma D21.5).
Linear sector (Theorem D21.3): $x=\Sigma\nabla\psi$ with zero solenoidal part, the $H^{-1}$ value $\beta+n$ , axis ratio and CMH quotient $1+n/\beta\le2$ , and the exact third-moment identity $\tau_Ze_1=e_1+\tfrac12T_3(e_1)Z$ , using steps 2–3 and Lemma D21.6.
Cube cone (Theorem D21.4): the kernel of the uniform law on $[-1,1]^{n-1}$ (Lemma D21.7, Lemma D21.8) gives the gate matrix (D21.34), whose eigenvalues are at most 2, hence (16.15); at $n=\beta=2$ the measure is a rotated product of exponentials.

Scope. This dossier proves, in the order stated, Proposition 17.1 (Theorem D21.2 below), Proposition 17.2 (Theorem D21.3) and Corollary 17.4 (Theorem D21.4), for the exponential cone measures of Definition 17.1. Every step is analytic. The only inputs from outside this file are: the compact-target regular moment-map theorem recorded as Theorem 4.1 (published; applied to the uniform probability on a convex body), the uniqueness-up-to-translation part of the moment-measure theorem of Cordero-Erausquin & Klartag, 2015, and the elementary moments of the Gamma law. Proposition 16.2 and Lemma 16.2 are not used; where the manuscript statements refer to them, this dossier proves the relevant identity directly and records the interpretation (§5. Remarks: hypotheses, conventions, and what is not claimed).

0. Standing conventions¶

Moment potentials. A convex function $\psi:\R^d\to\R\cup\{+\infty\}$ is essentially continuous (Definition 2 of Cordero-Erausquin & Klartag, 2015) if it is lower semi-continuous and its set of discontinuity points has zero $\mathcal H^{d-1}$ -measure. Following (4.1), a moment potential of a probability measure $\eta$ on $\R^d$ is an essentially-continuous convex function $\psi:\R^d\to\R\cup\{+\infty\}$ with $\int e^{-\psi}=1$ whose moment measure is $\eta$ , that is $(\nabla\psi)_\#(e^{-\psi}\dd y)=\eta$ . A finite convex function on $\R^d$ is continuous, hence essentially continuous; so every finite convex $\psi$ with $\int e^{-\psi}=1$ and $(\nabla\psi)_\#(e^{-\psi}\dd y)=\eta$ is a moment potential of $\eta$ in this sense, and in particular so is each of the potentials constructed in this dossier ( $\varphi$ of (D21.11), the rescaled $\lambda$ of (D21.8), the product $\Lambda$ of Lemma D21.8, and $\phi_T$ of Lemma D21.6). We use the following published fact.

This is Theorem 2 of the arXiv version 1304.0630v1 of Cordero-Erausquin & Klartag, 2015 (see also Klartag, 2014), stated there for a finite Borel measure $\mu$ with $0<\mu(\R^d)<\infty$ , not supported in a lower-dimensional subspace, with barycenter at the origin (so in particular with finite first moment); the conclusion there is the existence of an essentially-continuous convex $\psi:\R^d\to\R\cup\{+\infty\}$ with moment measure $\mu$ , unique up to translation. For a probability measure $\eta$ “not supported in a lower-dimensional subspace” is “not supported in a hyperplane”, and the normalization $\int e^{-\psi}=\mu(\R^d)=1$ is part of the definition of the moment measure. It is the “essentially unique” of (4.1). The theorem number refers to the arXiv text; the numbering in the journal version is noted in Remark D21.6(a).

The canonical Stein kernel. If $\eta$ has a moment potential $\psi\in C^\infty(\R^d)$ with $D^2\psi\succ0$ everywhere and $\nabla\psi$ a diffeomorphism of $\R^d$ onto an open set $\Omega$ , the canonical Stein kernel of $\eta$ is, as in (4.19),

\tau_\eta(x)=D^2\psi\bigl((\nabla\psi)^{-1}(x)\bigr),\qquad x\in\Omega.

(D21.1)

It does not depend on the translate: if $\tilde\psi(y)=\psi(y+a)$ then $\nabla\tilde\psi=\nabla\psi(\cdot+a)$ , $(\nabla\tilde\psi)^{-1}(x)=(\nabla\psi)^{-1}(x)-a$ , and $D^2\tilde\psi((\nabla\tilde\psi)^{-1}(x))=D^2\psi((\nabla\psi)^{-1}(x))$ . By Theorem D21.1 the kernel is therefore an invariant of $\eta$ , and any moment potential of $\eta$ with the regularity above computes it.

Weighted divergence and closed forms. Let $\Omega\subset\R^d$ be a connected open set and $\eta=\rho\dd x$ a probability measure with $\rho\in C^\infty(\Omega)$ , $\rho>0$ on $\Omega$ , $\rho=0$ off $\Omega$ . For a $C^1$ field $v$ on $\Omega$ , $\Div_\eta v=\rho^{-1}\Div(\rho v)$ as in (16.2). Let

\mathscr C=\R+C_c^\infty(\R^d)

(D21.2)

(functions on $\R^d$ , restricted to $\Omega$ ). For a continuous field $M$ of positive definite symmetric matrices on $\Omega$ with $\E_\eta\Tr M<\infty$ , the pre-form $\calE_M(f,g)=\E_\eta\inner{M\nabla f}{\nabla g}$ is finite on $\mathscr C$ .

Lemma D21.1 (Closability, kernel, operator)

In the setting above, $\calE_M$ on $\mathscr C$ is closable in $L^2(\eta)$ . Denote the closure again by $\calE_M$ , its domain by $\Dom(\calE_M)$ , and for $f\in\Dom(\calE_M)$ write $\nabla f$ for the $L^2(\eta;\R^d)$ -limit, weighted by $M^{1/2}$ , of the gradients of an approximating sequence, so that $\calE_M(f,g)=\E_\eta\inner{M\nabla f}{\nabla g}$ on the whole domain. The kernel of $\calE_M$ consists exactly of the constants. The nonnegative self-adjoint operator $\Aop_M$ of $\calE_M$ has domain

\Dom(\Aop_M)=\bigl\{g\in\Dom(\calE_M):\ \exists\,h\in L^2(\eta)\ \text{with}\ \calE_M(g,f)=\E_\eta[hf]\ \ \forall f\in\Dom(\calE_M)\bigr\},\qquad \Aop_Mg=h .

(D21.3)

Proof D21.1

Let $f_j\in\mathscr C$ with $f_j\to0$ in $L^2(\eta)$ and $M^{1/2}\nabla f_j\to v$ in $L^2(\eta;\R^d)$ . On a compact $Q\subset\Omega$ the density $\rho$ is bounded above and below and $M^{\pm1/2}$ are bounded, so $f_j\to0$ and $\nabla f_j\to M^{-1/2}v$ in $L^2(Q,\dd x)$ . For $\phi\in C_c^\infty(\operatorname{int}Q)$ , $\int\nabla f_j\,\phi=-\int f_j\nabla\phi\to0$ , hence $M^{-1/2}v=0$ a.e. on $Q$ ; exhausting $\Omega$ by compacts gives $v=0$ . This is closability. If $f\in\Dom(\calE_M)$ has $\calE_M(f)=0$ , the same local argument shows that the distributional gradient of $f$ on $\Omega$ vanishes, so $f$ is a.e. equal to a constant on the connected open set $\Omega$ ; conversely constants lie in $\mathscr C$ with zero energy. The description of the operator is Kato’s first representation theorem for closed nonnegative forms.

When $\Omega$ , $\eta$ are as above and $M$ is bounded near infinity in a suitable sense, the domain contains functions of polynomial growth; the instances needed are verified where they are used (Lemma D21.5). We write $\Aop_1=\Aop_\Sigma$ for the covariance generator of §The Hodge content: the affine channel and the solenoidal excess and $\Aop=\Aop_\tau$ for the Stein generator $-L_\mu$ of (16.2), both understood as the operators of the closures from $\mathscr C$ (the no-flux convention of §Conventions, domains, and affine covariance).

Polynomial growth. $\mathscr P(\R^d)$ denotes the class of $f\in C^1(\R^d)$ for which there are $C,p$ with $\abs{f(x)}+\abs{\nabla f(x)}\le C(1+\abs x)^p$ for all $x$ . Note $\mathscr C\subset\mathscr P$ .

1. The exponential cone measures¶

Fix $n\ge2$ , $m=n-1$ , a convex body $K\subset\R^m$ with barycenter at the origin, and $\beta\ge n$ . Write $x=(x_1,x')\in\R\times\R^m$ , let $C_K$ be the open cone of Definition 17.1, and let

\gamma_\beta(s)=\frac{s^{\beta-1}e^{-s}}{\Gamma(\beta)}\one_{s>0}, \qquad c_K=\sup_{u\in K}\abs{(1,u)}<\infty .

(D21.4)

For $x\in C_K$ put $s=x_1>0$ and $u=x'/x_1\in\operatorname{int}K$ , so that $x=s(1,u)$ .

Lemma D21.2 (Structure of

\mu_{K,\beta}

)

Let $S\sim\GammaLaw(\beta,1)$ and $U$ uniform on $K$ be independent, and $X=S(1,U)$ .

(a) $\int_{C_K}x_1^{\beta-n}e^{-x_1}\dd x=\Gamma(\beta)\abs K$ , and the law of $X$ is $\mu_{K,\beta}$ , with density $\rho(x)=\dfrac{x_1^{\beta-n}e^{-x_1}}{\Gamma(\beta)\abs K}\one_{C_K}(x)$ .

(b) $\mu_{K,\beta}$ is log-concave, and $\abs X\le c_KS$ , so $\mu_{K,\beta}$ has finite moments of all orders.

(c) $\E S^k=\beta(\beta+1)\cdots(\beta+k-1)$ for $k\in\mathbb N$ ; in particular $\E S=\beta$ , $\Var S=\beta$ , $\E S^2=\beta(\beta+1)$ , $\E S^3=\beta(\beta+1)(\beta+2)$ , $\E(S-\beta)^3=2\beta$ and $\E[(S-\beta)S^2]=2\beta(\beta+1)$ .

(d) $\E X=\beta e_1$ and $\Sigma:=\Cov(X)=\beta\,e_1e_1^\top\oplus\beta(\beta+1)\Cov(U)$ , which is positive definite; this is (17.24). Consequently $\Sigma e_1=\beta e_1$ , $\Sigma^{-1}=\beta^{-1}e_1e_1^\top\oplus(\beta(\beta+1))^{-1}\Cov(U)^{-1}$ , and $\Sigma^{\pm1/2}e_1=\beta^{\pm1/2}e_1$ .

Proof D21.2

(a) The map $\Psi:(s,u)\mapsto(s,su)$ is a diffeomorphism of $(0,\infty)\times\operatorname{int}K$ onto $C_K$ , with inverse $x\mapsto(x_1,x'/x_1)$ and Jacobian determinant $\det\left(\begin{smallmatrix}1&0\\u&s\Id_m\end{smallmatrix}\right)=s^m$ . Hence for measurable $F\ge0$ ,

\int_{C_K}F(x)\dd x=\int_{\operatorname{int}K}\int_0^\infty F(s,su)\,s^m\dd s\dd u .

(D21.5)

With $F(x)=x_1^{\beta-n}e^{-x_1}$ the inner integral is $\int_0^\infty s^{\beta-1}e^{-s}\dd s =\Gamma(\beta)$ , giving the normalization. The joint density of $(S,U)$ is $\gamma_\beta(s)\abs K^{-1}$ , so the density of $X=\Psi(S,U)$ at $x=(s,su)$ is $\gamma_\beta(s)\abs K^{-1}s^{-m}=x_1^{\beta-1-m}e^{-x_1}/(\Gamma(\beta)\abs K)$ , which is $\rho$ .

(b) $-\log\rho(x)=x_1-(\beta-n)\log x_1+\mathrm{const}$ on $C_K$ and $+\infty$ off $C_K$ . The cone $C_K$ is convex (it is the cone over the convex set $\{1\}\times\operatorname{int}K$ ), $x\mapsto x_1$ is linear, and $x\mapsto-(\beta-n)\log x_1$ is convex on $\{x_1>0\}$ because $\beta-n\ge0$ . The bound $\abs X=S\abs{(1,U)}\le c_KS$ and the finiteness of all moments of $S$ give the moment claim.

\E(S-\beta)^3=\E S^3-3\beta\E S^2+3\beta^2\E S-\beta^3 =(\beta^3+3\beta^2+2\beta)-3\beta^3-3\beta^2+3\beta^3-\beta^3=2\beta,

(D21.6)

\E[(S-\beta)S^2]=\E S^3-\beta\E S^2=\beta(\beta+1)(\beta+2)-\beta^2(\beta+1)=2\beta(\beta+1).

(D21.7)

(d) $\E X=\E S\,(1,\E U)=\beta e_1$ since the barycenter of $K$ is 0. Then $X-\beta e_1=(S-\beta,SU)$ and, by independence and $\E U=0$ : $\Var(S-\beta)=\beta$ ; $\E[(S-\beta)SU]=\E[(S-\beta)S]\,\E U=0$ ; $\E[S^2UU^\top]=\E S^2\,\E UU^\top=\beta(\beta+1)\Cov(U)$ . Since $K$ has nonempty interior, $\Cov(U)\succ0$ , so $\Sigma\succ0$ ; the block formulas for $\Sigma^{-1}$ and $\Sigma^{\pm1/2}$ follow from the block-diagonal form.

Throughout, $\mu=\mu_{K,\beta}$ , $\bar\mu=\bar\mu_{K,\beta}$ is the law of $\bar x=x-\beta e_1$ , and $\E$ denotes expectation for $x\sim\mu$ (equivalently $\bar x\sim\bar\mu$ ). The translation $x\mapsto\bar x$ maps $C_K$ onto $C_K-\beta e_1$ and preserves gradients, weighted divergences, the classes $\mathscr C$ , $\mathscr P$ , and the closed forms of Lemma D21.1; we therefore state functional identities in the coordinate $x\in C_K$ and read them for $\bar\mu$ without further comment.

2. The cone lift of the moment map¶

Lemma D21.3 (The base and its rescaling)

Let $\lambda_K$ be a moment potential of the uniform probability on $K$ . Then $\lambda_K$ is smooth and strictly convex, $D^2\lambda_K\succ0$ everywhere, $\nabla\lambda_K$ is a diffeomorphism of $\R^m$ onto $\operatorname{int}K$ , the canonical kernel $\tau_K=D^2\lambda_K\circ(\nabla\lambda_K)^{-1}$ is smooth and positive definite on $\operatorname{int}K$ , and $\E\,\tau_K(U)=\Cov(U)$ for $U$ uniform on $K$ . Moreover, for $\beta>0$ , the function

\lambda(y')=\lambda_K(\beta y')-m\log\beta

(D21.8)

is a moment potential of the uniform probability on $\beta K$ , every moment potential of that law is a translate of it, and the canonical kernels satisfy

\tau_{\beta K}(\beta u)=\beta^2\,\tau_K(u),\qquad u\in\operatorname{int}K .

(D21.9)

Proof D21.3

The uniform probability on $K$ is $g\one_{\operatorname{int}K}\dd x$ with $g\equiv\abs K^{-1}$ , smooth and positive, and is centered because the barycenter of $K$ is 0. Theorem 4.1, applied in dimension $m$ with $P=K$ , gives a moment potential $\lambda_K^0$ of this law (the canonical one of that theorem) which is smooth and strictly convex, with $\nabla\lambda_K^0$ a diffeomorphism of $\R^m$ onto $\operatorname{int}K$ , and with $\E\tau_K(U)=\Cov(U)$ for the kernel $D^2\lambda_K^0\circ(\nabla\lambda_K^0)^{-1}$ . The law is centered, has finite first moment (it is compactly supported) and is not supported in a hyperplane ( $\operatorname{int}K$ is open and nonempty), so by Theorem D21.1 the given moment potential is a translate, $\lambda_K=\lambda_K^0(\cdot+a)$ ; translation preserves smoothness, strict convexity, the diffeomorphism property of the gradient (with the same image) and, by the invariance noted after (D21.1), the kernel, so every statement holds for $\lambda_K$ . The Jacobian of the diffeomorphism $\nabla\lambda_K$ is $D^2\lambda_K$ , which is therefore invertible; being positive semidefinite by convexity it is positive definite, and $\tau_K$ is smooth and positive definite as the composition of $D^2\lambda_K$ with the smooth inverse.

For (D21.8): by the substitution $z=\beta y'$ , $\int e^{-\lambda(y')}\dd y'=\beta^m\int e^{-\lambda_K(\beta y')}\dd y'=\int e^{-\lambda_K(z)}\dd z=1$ , and $e^{-\lambda(y')}\dd y'$ is the law of $Y_0/\beta$ when $Y_0\sim e^{-\lambda_K(z)}\dd z$ . Since $\nabla\lambda(y')=\beta(\nabla\lambda_K)(\beta y')$ , the pushforward of this law under $\nabla\lambda$ is the law of $\beta\nabla\lambda_K(Y_0)$ , which is the uniform probability on $\beta K$ because $\nabla\lambda_K(Y_0)$ is uniform on $K$ . The function $\lambda$ is finite, smooth and strictly convex, so by Theorem D21.1 (the uniform probability on $\beta K$ is centered, has finite first moment, and is not supported in a hyperplane) every moment potential of the uniform probability on $\beta K$ is a translate of $\lambda$ . In particular $\tau_{\beta K}$ may be computed from $\lambda$ : $D^2\lambda(y')=\beta^2D^2\lambda_K(\beta y')$ and $(\nabla\lambda)^{-1}(z)=\beta^{-1}(\nabla\lambda_K)^{-1}(z/\beta)$ , so

\tau_{\beta K}(z)=\beta^2D^2\lambda_K\bigl((\nabla\lambda_K)^{-1}(z/\beta)\bigr)=\beta^2\tau_K(z/\beta),

(D21.10)

which is (D21.9) at $z=\beta u$ .

From now on $\lambda$ denotes a moment potential of the uniform probability on $\beta K$ (any one; all are translates of (D21.8)); it is smooth, strictly convex with $D^2\lambda\succ0$ , and $\nabla\lambda:\R^m\to\beta\operatorname{int}K$ is a diffeomorphism. Set, for $y=(y_1,y')\in\R\times\R^m$ ,

E(y)=\exp\bigl(y_1+\lambda(y')/\beta\bigr),\qquad u(y')=\nabla\lambda(y')/\beta\in\operatorname{int}K,\qquad \varphi(y)=E(y)-\beta y_1+\log\Gamma(\beta).

(D21.11)

Theorem D21.2 (Cone lift of the moment map; Proposition 17.1)

(i) $\varphi\in C^\infty(\R^n)$ , $D^2\varphi\succ0$ everywhere, and

\nabla\varphi(y)=\bigl(E-\beta,\ E\,u\bigr),\qquad D^2\varphi(y)=E\begin{pmatrix}1&u^\top\\ u&uu^\top+D^2\lambda(y')/\beta\end{pmatrix},

(D21.12)

with $E=E(y)$ , $u=u(y')$ .

(ii) $\nabla\varphi$ is a diffeomorphism of $\R^n$ onto $C_K-\beta e_1$ ; explicitly $\nabla\varphi(y)=x-\beta e_1$ with $x=(x_1,x_1u)$ , $x_1=E(y)$ , $u=u(y')$ .

(iii) $\int_{\R^n}e^{-\varphi}=1$ and $(\nabla\varphi)_\#(e^{-\varphi}\dd y)=\bar\mu$ . Hence $\varphi$ is a moment potential of $\bar\mu$ , and every moment potential of $\bar\mu$ is a translate of $\varphi$ .

(iv) The canonical Stein kernel of $\bar\mu$ at $\bar x=x-\beta e_1$ , $x=(x_1,x_1u)\in C_K$ , is

\tau(\bar x)=x_1\begin{pmatrix}1&u^\top\\ u&uu^\top+\beta\,\tau_K(u)\end{pmatrix},

(D21.13)

and in particular $\tau(\bar x)e_1=x$ . Moreover $\E\,\tau=\Sigma$ .

Proof D21.4

(i) Put $g(y)=y_1+\lambda(y')/\beta$ , a smooth convex function with $\nabla g=(1,u)$ and $D^2g=0\oplus D^2\lambda/\beta$ . Then $\varphi=e^g-\beta y_1+\log\Gamma(\beta)$ is smooth, $\nabla\varphi=e^g\nabla g-\beta e_1=(E-\beta,Eu)$ , and $D^2\varphi=e^g(\nabla g\nabla g^\top+D^2g)$ , which is (D21.12). For $v=(v_1,v')\ne0$ , $v^\top D^2\varphi\,v=E\bigl[(v_1+\inner u{v'})^2+v'^\top D^2\lambda\,v'/\beta\bigr]$ ; if this vanishes then $v'=0$ because $D^2\lambda\succ0$ , and then $v_1=0$ . So $D^2\varphi\succ0$ .

(ii) The map $\Theta:y\mapsto(E(y),u(y'))$ is a diffeomorphism of $\R^n$ onto $(0,\infty)\times\operatorname{int}K$ : it is smooth, and its inverse $(x_1,u)\mapsto\bigl(\log x_1-\lambda(y')/\beta,\ y'\bigr)$ with $y'=(\nabla\lambda)^{-1}(\beta u)$ is smooth because $\nabla\lambda$ is a diffeomorphism onto $\beta\operatorname{int}K$ . Composing with the diffeomorphism $\Psi$ of Lemma D21.2(a) and the translation by $-\beta e_1$ gives $\nabla\varphi=(\Psi\circ\Theta)-\beta e_1$ , a diffeomorphism of $\R^n$ onto $C_K-\beta e_1$ .

(iii) Write $\Phi(y')=e^{\lambda(y')/\beta}$ , so $E(y)=e^{y_1}\Phi(y')$ and $e^{-\varphi(y)}=\Gamma(\beta)^{-1}\exp(-E+\beta y_1)$ . Fix $y'$ and substitute $x_1=e^{y_1}\Phi(y')$ in the $y_1$ -integral: $\dd y_1=\dd x_1/x_1$ and $e^{\beta y_1}=x_1^\beta\Phi(y')^{-\beta}$ , so

e^{-\varphi(y)}\dd y_1 =\Gamma(\beta)^{-1}\Phi(y')^{-\beta}x_1^{\beta-1}e^{-x_1}\dd x_1 =e^{-\lambda(y')}\gamma_\beta(x_1)\dd x_1 .

(D21.14)

Therefore, by Tonelli, for every measurable $F\ge0$ on $(0,\infty)\times\operatorname{int}K$ ,

\int_{\R^n}F\bigl(\Theta(y)\bigr)e^{-\varphi(y)}\dd y =\int_{\R^m}e^{-\lambda(y')}\Bigl(\int_0^\infty F\bigl(x_1,u(y')\bigr)\gamma_\beta(x_1)\dd x_1\Bigr)\dd y' =\E\bigl[F(S,U)\bigr],

(D21.15)

where the last equality uses that $e^{-\lambda(y')}\dd y'$ is a probability measure whose pushforward under $\nabla\lambda$ is uniform on $\beta K$ , so that its pushforward under $u=\nabla\lambda/\beta$ is uniform on $K$ , and $S\sim\GammaLaw(\beta,1)$ is independent of $U$ . Taking $F\equiv1$ gives $\int e^{-\varphi}=1$ ; taking $F=G\circ\Psi$ for bounded measurable $G\ge0$ on $C_K$ gives $\int G(\Psi\Theta(y))e^{-\varphi}\dd y=\E\,G(X)$ , i.e.\ $(\Psi\circ\Theta)_\#(e^{-\varphi}\dd y)=\mu$ , and translating, $(\nabla\varphi)_\#(e^{-\varphi}\dd y)=\bar\mu$ . So $\varphi$ is a finite smooth convex moment potential of $\bar\mu$ . The measure $\bar\mu$ is centered (Lemma D21.2(d)), has finite first moment (Lemma D21.2(b)), and is not supported in a hyperplane because $C_K-\beta e_1$ is open and nonempty; Theorem D21.1 gives that every moment potential of $\bar\mu$ is a translate of $\varphi$ .

(iv) By (ii), (iii) and the convention (D21.1), the canonical kernel of $\bar\mu$ at $\bar x=\nabla\varphi(y)$ is $D^2\varphi(y)$ . In (D21.12), $E(y)=x_1$ , $u(y')=u$ , and $D^2\lambda(y')=\tau_{\beta K}(\nabla\lambda(y'))=\tau_{\beta K}(\beta u) =\beta^2\tau_K(u)$ by the definition (D21.1) of $\tau_{\beta K}$ and (D21.9). Substituting gives (D21.13); the first column is $x_1(1,u)=x$ . Finally, by independence of $x_1=S$ and $u=U$ under $\mu$ (Lemma D21.2), $\E[x_1]=\beta$ , $\E[x_1u]=\beta\E U=0$ and $\E\bigl[x_1(uu^\top+\beta\tau_K(u))\bigr]=\beta\bigl(\Cov(U)+\beta\Cov(U)\bigr) =\beta(\beta+1)\Cov(U)$ , using $\E\tau_K(U)=\Cov(U)$ from Lemma D21.3. This is $\Sigma$ .

The next lemma is the Stein identity for $\tau$ on a class large enough for all subsequent uses; it is the target-coordinate reading of (4.20) for the noncompact target $C_K$ .

Proof D21.5

(a) Since $\tau_K(u)\succeq0$ , $\abs{(\tau_K)_{ij}}\le\Tr\tau_K$ , and $\abs u\le c_K$ on $K$ , (D21.13) gives $\abs{\tau_{ij}}\le x_1\bigl(1+c_K^2+\beta\Tr\tau_K(u)\bigr)$ . By independence of $x_1$ and $u$ , $\E[x_1^q\abs{\tau_{ij}}]\le\E[x_1^{q+1}]\,(1+c_K^2+\beta\Tr\Cov(U))<\infty$ , using $\E\Tr\tau_K(U)=\Tr\Cov(U)$ . The second claim follows from $\abs x\le c_Kx_1$ .

(b) Let $F=f\circ\nabla\varphi\in C^1(\R^n)$ . By the chain rule and $\tau_{ij}(\nabla\varphi(y))=\varphi_{ij}(y)$ , $\partial_iF(y)=\sum_j\partial_jf(\nabla\varphi(y))\,\tau_{ij}(\nabla\varphi(y))$ . By the pushforward of Theorem D21.2(iii),

\int\partial_i\varphi\,F\,e^{-\varphi}\dd y=\E[\bar x_if(\bar x)],\qquad \int\partial_iF\,e^{-\varphi}\dd y=\E\bigl[\inner{\tau e_i}{\nabla f}\bigr],\qquad \int\abs F\,e^{-\varphi}\dd y=\E\abs{f(\bar x)},

(D21.18)

and all three are absolutely convergent: $\bar\mu$ has all moments, $f$ and $\nabla f$ have polynomial growth, and (a) bounds $\E[(1+\abs x)^p\abs{\tau_{ij}}]$ . Choose $\chi\in C_c^\infty(\R^n)$ with $0\le\chi\le1$ , $\chi=1$ on the unit ball, and set $\chi_R=\chi(\cdot/R)$ , so $\norm{\nabla\chi_R}_\infty\le\norm{\nabla\chi}_\infty/R$ . Since $F\chi_R$ has compact support, integration by parts on $\R^n$ gives

\int\partial_i\varphi\,F\chi_R\,e^{-\varphi} =-\int\partial_i(e^{-\varphi})F\chi_R =\int e^{-\varphi}\bigl(\partial_iF\,\chi_R+F\,\partial_i\chi_R\bigr).

(D21.19)

As $R\to\infty$ the left side and the first term on the right converge by dominated convergence to $\E[\bar x_if]$ and $\E\inner{\tau e_i}{\nabla f}$ respectively, and the last term is bounded by $\norm{\nabla\chi}_\infty R^{-1}\E\abs f\to0$ . This is (D21.16).

(c) Take $i=1$ in (b): $\tau e_1=x$ by Theorem D21.2(iv), $\bar x_1=x_1-\beta$ , and $f(\bar x)$ ranges over $\mathscr P$ as $f$ does, so (D21.17) is (D21.16) written in the coordinate $x$ .

Remark D21.1 (Direct radial proof of (D21.17))

The first-column identity also follows from a one-dimensional integration by parts that makes the zero boundary flux visible. For $x=s(1,u)$ and $f\in\mathscr P$ , $\frac{\dd}{\dd s}f(s(1,u))=\inner{(1,u)}{\nabla f(s(1,u))}$ , so $\inner{x}{\nabla f(x)}=s\,\frac{\dd}{\dd s}f(s(1,u))$ . Since $(s\gamma_\beta)'(s)=(\beta-s)\gamma_\beta(s)$ and $s\gamma_\beta(s)f(s(1,u))\to0$ as $s\downarrow0$ (as $s^\beta$ ) and as $s\to\infty$ (as $s^{\beta+p}e^{-s}$ ), for each fixed $u\in\operatorname{int}K$ ,

\int_0^\infty s\,\gamma_\beta(s)\,\frac{\dd}{\dd s}f(s(1,u))\dd s =\int_0^\infty(s-\beta)\gamma_\beta(s)f(s(1,u))\dd s ,

(D21.20)

both integrals being absolutely convergent. Integrating in $u$ against the uniform probability on $K$ and using Lemma D21.2(a) gives (D21.17). Geometrically, the field $x$ is tangent to the lateral boundary of the dilation-invariant cone $C_K$ , and $\rho\abs x\to0$ at the apex and at infinity, so no boundary term appears even for test functions that do not vanish on $\partial C_K$ ; this is the weak zero-flux formulation of Theorem 4.1.

3. The linear sector¶

Let $\Omega=C_K$ , $\eta=\mu$ , and apply Lemma D21.1 with $M=\Sigma$ (constant) and with $M=\tau$ (smooth and positive definite on $C_K$ by Theorem D21.2, with $\E\Tr\tau=\Tr\Sigma<\infty$ ). This defines the closed covariance form $\calE_\Sigma$ with operator $\Aop_1$ , and the closed Stein form $\calE_\tau$ with operator $\Aop=-L_\mu$ . Set

\psi(x)=\tfrac12\,x^\top\Sigma^{-1}x,\qquad\text{so that}\qquad \nabla\psi=\Sigma^{-1}x, \quad \Sigma\nabla\psi=x .

(D21.21)

Proof D21.6

(a) Let $f$ be a polynomial and $\chi_R$ as in Lemma D21.4, so $\chi_Rf\in C_c^\infty(\R^n)\subset\mathscr C$ . Then $\chi_Rf\to f$ in $L^2(\mu)$ by dominated convergence ( $\mu$ has all moments), and $\nabla(\chi_Rf)-\nabla f=(\chi_R-1)\nabla f+f\nabla\chi_R$ . For $M\in\{\Sigma,\tau\}$ ,

\E\inner{M(\nabla(\chi_Rf)-\nabla f)}{\nabla(\chi_Rf)-\nabla f} \le2\,\E\bigl[(1-\chi_R)^2\inner{M\nabla f}{\nabla f}\bigr] +\frac{2\norm{\nabla\chi}_\infty^2}{R^2}\,\E\bigl[f^2\,\Tr M\bigr],

(D21.22)

using $\inner{Mv}{v}\le\Tr M\,\abs v^2$ . Both expectations are finite by Lemma D21.4(a) (for $M=\tau$ ) or because $\Sigma$ is constant, so the first term tends to 0 by dominated convergence and the second is $O(R^{-2})$ . Hence $(\chi_Rf)$ is Cauchy in the form norm, $f$ lies in the closed domain, and its form-gradient is the limit $\nabla f$ .

(b) By (a), $\psi\in\Dom(\calE_\Sigma)$ with gradient $\Sigma^{-1}x$ . For $f\in\mathscr C\subset\mathscr P$ , (D21.17) gives $\calE_\Sigma(\psi,f)=\E\inner{\Sigma\nabla\psi}{\nabla f}=\E\inner x{\nabla f}=\E[\bar x_1f]$ . Both sides are continuous in $f$ for the form norm (the left because $\psi\in\Dom(\calE_\Sigma)$ , the right because $\bar x_1\in L^2(\mu)$ ), and $\mathscr C$ is dense in $\Dom(\calE_\Sigma)$ by construction, so the identity holds for all $f\in\Dom(\calE_\Sigma)$ . By Lemma D21.1 this says $\psi\in\Dom(\Aop_1)$ with $\Aop_1\psi=\bar x_1$ . This also proves (d).

(c) By (a), $g\in\Dom(\calE_\tau)$ with gradient $e_1$ , and for $f\in\mathscr C$ , $\calE_\tau(g,f)=\E\inner{\tau e_1}{\nabla f}=\E\inner x{\nabla f}=\E[\bar x_1f]$ by Theorem D21.2(iv) and (D21.17). The same continuity and density argument gives $g\in\Dom(\Aop)$ , $\Aop g=\bar x_1$ . Since $g$ is not constant it is not in $\ker\Aop$ (Lemma D21.1).

Proof D21.7

$\phi_T$ is smooth with $\nabla\phi_T(y)=T\nabla\phi(T^\top y)$ and $D^2\phi_T(y)=TD^2\phi(T^\top y)T^\top\succ0$ ; $\nabla\phi_T=T\circ\nabla\phi\circ T^\top$ is a diffeomorphism of $\R^d$ onto $T\Omega$ . If $Y_0\sim e^{-\phi(z)}\dd z$ then $W=(T^\top)^{-1}Y_0$ has density $e^{-\phi(T^\top y)}\abs{\det T}=e^{-\phi_T(y)}$ , so $\int e^{-\phi_T}=1$ , and $\nabla\phi_T(W)=T\nabla\phi(Y_0)\sim T_\#\eta$ . Thus $\phi_T$ is a moment potential of $T_\#\eta$ . For the kernel, $(\nabla\phi_T)^{-1}(Tx)=(T^\top)^{-1}(\nabla\phi)^{-1}(x)$ , hence $\tau_{T_\#\eta}(Tx)=D^2\phi_T\bigl((T^\top)^{-1}(\nabla\phi)^{-1}(x)\bigr) =TD^2\phi\bigl((\nabla\phi)^{-1}(x)\bigr)T^\top=T\tau_\eta(x)T^\top$ . (If $T_\#\eta$ satisfies the hypotheses of Theorem D21.1 — it does whenever $\eta$ does — this is the canonical kernel of $T_\#\eta$ .)

Theorem D21.3 (The linear sector; Proposition 17.2)

Let $\mu=\bar\mu_{K,\beta}$ , with $\Sigma$ and $\tau$ as in Lemma D21.2 and Theorem D21.2, and $\bar x=x-\beta e_1$ .

(i) The field $x=\tau e_1$ satisfies $\Div_\mu x=-\bar x_1$ pointwise on $C_K$ , and the weak zero-flux identity $\E\inner x{\nabla f}=\E[\bar x_1f]$ holds for every $f\in\mathscr P(\R^n)$ and every $f\in\Dom(\calE_\Sigma)$ . It is a covariance gradient, $x=\Sigma\nabla\psi$ with $\psi$ as in (D21.21), $\psi\in\Dom(\Aop_1)$ and $\Aop_1\psi=\bar x_1$ ; in the decomposition of Proposition 16.2 applied to the field $u=x$ (with $h=\bar x_1$ ), the solenoidal part is $w=x-\Sigma\nabla\Aop_1^{-1}\bar x_1=0$ . Moreover

\sup_{f\in\Dom(\calE_\Sigma)\setminus\R} \frac{\bigl(\E[\bar x_1f]\bigr)^2}{\E\inner{\Sigma\nabla f}{\nabla f}} =\E\bigl[x^\top\Sigma^{-1}x\bigr]=\beta+n,

(D21.23)

and the supremum is attained at $f=\psi$ . The same value and attainment hold with the supremum restricted to nonconstant $f\in\mathscr P(\R^n)$ .

(ii)

\frac{e_1^\top\,\E[\tau\Sigma^{-1}\tau]\,e_1}{e_1^\top\Sigma e_1}=1+\frac n\beta\le2,

(D21.24)

with equality if and only if $\beta=n$ . The function $g(x)=x_1$ lies in $\Dom(\Aop)\setminus\ker\Aop$ , and the CMH Rayleigh quotient (16.5) at $g$ equals

\frac{\E\inner{\tau\nabla g}{\Sigma^{-1}\tau\nabla g}}{\E(L_\mu g)^2} =\frac{\beta+n}{\beta}=1+\frac n\beta .

(D21.25)

(iii) Let $Z=\Sigma^{-1/2}\bar x$ and $\tau_Z(Z)=\Sigma^{-1/2}\tau(\bar x)\Sigma^{-1/2}$ , the canonical kernel of the isotropic law of $Z$ . Then, with $T_3(e_1)=\E[Z_1\,Z\otimes Z]$ ,

T_3(e_1)=2\beta^{-1/2}\,\Id_n,\qquad \tau_Ze_1=e_1+\beta^{-1/2}Z=e_1+\tfrac12T_3(e_1)Z\quad\text{identically},

(D21.26)

so $\norm{T_3(e_1)}_{\HS}^2=4n/\beta$ , and the residual $v_{e_1}:=\tau_Ze_1-e_1-\tfrac12T_3(e_1)Z$ vanishes identically for every base $K$ and every $\beta\ge n$ .

In particular, for $\beta=n$ : $e_1^\top\E[\tau\Sigma^{-1}\tau]e_1=2\,e_1^\top\Sigma e_1$ , which is equality in (16.15) in the direction $e_1$ , and $\norm{T_3(e_1)}_{\HS}=2$ .

Proof D21.8

(i) On $C_K$ , with $\rho$ the density of Lemma D21.2(a), $\nabla\log\rho=\bigl((\beta-n)/x_1-1\bigr)e_1$ and $\Div x=n$ , so

\Div_\mu x=\Div x+\inner{x}{\nabla\log\rho}=n+(\beta-n)-x_1=\beta-x_1=-\bar x_1 .

(D21.27)

The weak identity on $\mathscr P$ is (D21.17), and on $\Dom(\calE_\Sigma)$ it is Lemma D21.5(d). That $x=\Sigma\nabla\psi$ is (D21.21); $\psi\in\Dom(\Aop_1)$ with $\Aop_1\psi=\bar x_1$ is Lemma D21.5(b). Since $\ker\Aop_1$ consists of the constants (Lemma D21.1), $L^2_0(\mu)=(\ker\Aop_1)^\perp$ is the space of centered functions, $\bar x_1\in L^2_0(\mu)$ , and $\Aop_1$ is injective on $\Dom(\Aop_1)\cap L^2_0(\mu)$ ; the unique centered solution of $\Aop_1\psi_0=\bar x_1$ is $\psi_0=\psi-\E\psi$ , so $\Aop_1^{-1}\bar x_1=\psi-\E\psi$ and $w=x-\Sigma\nabla(\psi-\E\psi)=x-x=0$ .

For (D21.23): for $f\in\Dom(\calE_\Sigma)$ , Lemma D21.5(d) and the Cauchy–Schwarz inequality for the nonnegative form $\calE_\Sigma$ give

\bigl(\E[\bar x_1f]\bigr)^2=\calE_\Sigma(\psi,f)^2\le\calE_\Sigma(\psi)\,\calE_\Sigma(f), \qquad \calE_\Sigma(\psi)=\E\inner{\Sigma\nabla\psi}{\nabla\psi}=\E\bigl[x^\top\Sigma^{-1}x\bigr],

(D21.28)

and $\calE_\Sigma(f)>0$ for nonconstant $f$ . At $f=\psi$ the ratio equals $\calE_\Sigma(\psi)^2/\calE_\Sigma(\psi)=\calE_\Sigma(\psi)$ . The same argument applies verbatim on $\mathscr P$ , using (D21.17) and the pointwise Cauchy–Schwarz inequality $\inner{x}{\nabla f}=\inner{\Sigma^{-1/2}x}{\Sigma^{1/2}\nabla f}$ ; $\psi\in\mathscr P$ . For the value, by Lemma D21.2(c),(d) and independence of $x_1$ and $u$ ,

\begin{aligned} \E\bigl[x^\top\Sigma^{-1}x\bigr] &=\E\Bigl[\frac{x_1^2}{\beta}+\frac{x_1^2\,u^\top\Cov(U)^{-1}u}{\beta(\beta+1)}\Bigr] =\frac{\E x_1^2}{\beta}+\frac{\E x_1^2}{\beta(\beta+1)}\,\E\Tr\bigl(\Cov(U)^{-1}UU^\top\bigr)\\ &=(\beta+1)+\Tr\bigl(\Cov(U)^{-1}\Cov(U)\bigr)=(\beta+1)+m=\beta+n . \end{aligned}

(D21.29)

(ii) Since $\tau$ is symmetric, $e_1^\top\tau\Sigma^{-1}\tau e_1=(\tau e_1)^\top\Sigma^{-1}(\tau e_1) =x^\top\Sigma^{-1}x$ , a nonnegative random variable with expectation $\beta+n$ by (i); and $e_1^\top\Sigma e_1=\beta$ . Hence the ratio is $(\beta+n)/\beta=1+n/\beta$ , which is $\le2$ if and only if $n\le\beta$ , with equality if and only if $\beta=n$ . For the Rayleigh quotient, Lemma D21.5(c) gives $g\in\Dom(\Aop)\setminus\ker\Aop$ with form-gradient $\nabla g=e_1$ and $L_\mu g=-\Aop g=-\bar x_1$ , so the numerator of (16.5) is $\E\inner{\tau e_1}{\Sigma^{-1}\tau e_1}=\beta+n$ and the denominator is $\E\bar x_1^2=\Var(x_1)=\beta$ .

(iii) By Lemma D21.2(d) the law of $Z$ is isotropic, and by Lemma D21.6 with $T=\Sigma^{-1/2}$ its canonical kernel at $Z$ is $\Sigma^{-1/2}\tau(\bar x)\Sigma^{-1/2}$ . Using $\Sigma^{-1/2}e_1=\beta^{-1/2}e_1$ , $\tau e_1=x=\bar x+\beta e_1$ and $\Sigma^{-1/2}\bar x=Z$ ,

\tau_Ze_1=\beta^{-1/2}\Sigma^{-1/2}\tau e_1 =\beta^{-1/2}\Sigma^{-1/2}(\bar x+\beta e_1) =\beta^{-1/2}Z+\beta^{1/2}\cdot\beta^{-1/2}e_1=e_1+\beta^{-1/2}Z .

(D21.30)

Next, $Z_1=\bar x_1/\sqrt\beta=(S-\beta)/\sqrt\beta$ and $Z'=(\beta(\beta+1))^{-1/2}\Cov(U)^{-1/2}SU$ . By independence and Lemma D21.2(c):

\begin{aligned} \E[Z_1^3]&=\beta^{-3/2}\E(S-\beta)^3=2\beta^{-1/2},\\ \E[Z_1^2Z']&=\beta^{-1}(\beta(\beta+1))^{-1/2}\,\E[(S-\beta)^2S]\;\Cov(U)^{-1/2}\E U=0,\\ \E[Z_1Z'Z'^\top]&=\frac{\E[(S-\beta)S^2]}{\sqrt\beta\,\beta(\beta+1)}\,\Cov(U)^{-1/2}\E[UU^\top]\Cov(U)^{-1/2} =\frac{2\beta(\beta+1)}{\sqrt\beta\,\beta(\beta+1)}\,\Id_m=2\beta^{-1/2}\Id_m . \end{aligned}

(D21.31)

Hence $T_3(e_1)=\E[Z_1ZZ^\top]=2\beta^{-1/2}\Id_n$ , $\norm{T_3(e_1)}_{\HS}^2=4\beta^{-1}n$ , $\tfrac12T_3(e_1)Z=\beta^{-1/2}Z$ , and the displayed identity for $\tau_Ze_1$ shows $v_{e_1}\equiv0$ .

The final sentence is (ii) and (iii) at $\beta=n$ : $1+n/\beta=2$ and $4n/\beta=4$ .

4. The cube cone¶

Proof D21.9

By Lemma D21.3 in dimension $m=1$ with $K=[-1,1]$ , a moment potential $\lambda_1$ is smooth, $\lambda_1''>0$ , and $\lambda_1'$ is an increasing diffeomorphism of $\R$ onto $(-1,1)$ . For bounded measurable $h$ on $(-1,1)$ the defining pushforward and the substitution $t=\lambda_1'(y)$ , $\dd t=\lambda_1''(y)\dd y$ , give

\int_\R h(\lambda_1'(y))e^{-\lambda_1(y)}\dd y=\int_{-1}^1h(t)\tfrac12\dd t =\int_\R h(\lambda_1'(y))\tfrac12\lambda_1''(y)\dd y .

(D21.32)

Since $h\circ\lambda_1'$ ranges over all bounded measurable functions on $\R$ , the continuous functions $e^{-\lambda_1}$ and $\tfrac12\lambda_1''$ coincide: $\lambda_1''=2e^{-\lambda_1}$ (the one-dimensional case of (4.3)). Consequently

\frac{\dd}{\dd y}\Bigl[(\lambda_1')^2+4e^{-\lambda_1}\Bigr] =2\lambda_1'\bigl(\lambda_1''-2e^{-\lambda_1}\bigr)=0,

(D21.33)

so $(\lambda_1')^2+4e^{-\lambda_1}\equiv c$ . As $y\to+\infty$ , $\lambda_1'(y)\to1$ ; moreover $\lambda_1'(y_0)>0$ for some $y_0$ , so $\lambda_1(y)\ge\lambda_1(y_0)+\lambda_1'(y_0)(y-y_0)\to\infty$ and $e^{-\lambda_1(y)}\to0$ . Thus $c=1$ and $2e^{-\lambda_1}=\tfrac12\bigl(1-(\lambda_1')^2\bigr)$ . At $t=\lambda_1'(y)$ the kernel is $\tau_1(t)=\lambda_1''(y)=2e^{-\lambda_1(y)}=\tfrac12(1-t^2)$ .

This agrees with (17.1): for $\rho\equiv\tfrac12$ on $(-1,1)$ , $-\rho(t)^{-1}\int_{-1}^tv\rho(v)\dd v=\tfrac12(1-t^2)$ .

Proof D21.10

Let $\lambda_1$ be as in Lemma D21.7 and $\Lambda(y')=\sum_{j=1}^m\lambda_1(y_j)$ . Then $\Lambda$ is smooth and strictly convex, $\int e^{-\Lambda}=\prod_j\int e^{-\lambda_1}=1$ , $e^{-\Lambda}\dd y'$ is the product of $m$ copies of $e^{-\lambda_1}\dd y_j$ , and $\nabla\Lambda(y')=(\lambda_1'(y_j))_j$ pushes it to the product of $m$ copies of the uniform probability on $[-1,1]$ , i.e. to the uniform probability on $K$ ; and $\nabla\Lambda$ is a diffeomorphism onto $(-1,1)^m=\operatorname{int}K$ . So $\Lambda$ is a moment potential of the uniform law on $K$ with the regularity of (D21.1), and by Theorem D21.1 it computes the canonical kernel: $D^2\Lambda=\diag(\lambda_1''(y_j))$ and $(\nabla\Lambda)^{-1}(u)=((\lambda_1')^{-1}(u_j))_j$ , so $\tau_K(u)=\diag(\tau_1(u_j))=\tfrac12\diag(1-u_j^2)$ . Finally the coordinates of $U$ are i.i.d. uniform on $[-1,1]$ with $\E U_j^2=\int_{-1}^1t^2\tfrac12\dd t=\tfrac13$ .

Theorem D21.4 (The cube cone; Corollary 17.4)

Let $K=[-1,1]^{n-1}$ , $n\ge2$ , $\beta\ge n$ , and $\mu=\bar\mu_{K,\beta}$ . Then $\tau_K(u)=\tfrac12\diag(1-u_j^2)$ , $\E[\tau\Sigma^{-1}\tau]$ is finite, and the normalized gate matrix satisfies

\Sigma^{-1/2}\,\E[\tau\Sigma^{-1}\tau]\,\Sigma^{-1/2} =\Bigl(1+\frac n\beta\Bigr)\ \oplus\ G'\,\Id_{n-1}, \qquad G'=\frac{6\beta^2+11\beta+5n+4}{5\beta(\beta+1)} .

(D21.34)

Both eigenvalues are at most 2: $1+n/\beta\le2$ with equality if and only if $\beta=n$ , and $G'\le2$ with equality if and only if $n=\beta=2$ . Hence $\E[\tau\Sigma^{-1}\tau]\preceq2\Sigma$ , which is (16.15) for $\mu$ , for every $n\ge2$ and $\beta\ge n$ . For $n=\beta=2$ the matrix (D21.34) is $2\,\Id_2$ , and $\mu$ is the image of a product of two centered standard exponential laws under $\sqrt2$ times an orthogonal map; equivalently, in the orthonormal frame $(e_1\pm e_2)/\sqrt2$ , $\mu$ is a product of two centered exponential laws.

Proof D21.11

The kernel of the base is Lemma D21.8; write $D=\diag(1-u_j^2)$ and $B=uu^\top+\tfrac\beta2D$ , so that by (D21.13) $\tau=x_1\left(\begin{smallmatrix}1&u^\top\\u&B\end{smallmatrix}\right)$ . Since $\Cov(U)=\tfrac13\Id_m$ , Lemma D21.2(d) gives $\Sigma=\beta\oplus\sigma^2\Id_m$ with $\sigma^2=\beta(\beta+1)/3$ , and $\Sigma^{-1}=\beta^{-1}\oplus\sigma^{-2}\Id_m$ . Multiplying out,

\tau\Sigma^{-1}\tau =x_1^2\begin{pmatrix} \dfrac1\beta+\dfrac{\abs u^2}{\sigma^2} & \dfrac{u^\top}{\beta}+\dfrac{u^\top B}{\sigma^2}\\ \dfrac u\beta+\dfrac{Bu}{\sigma^2} & \dfrac{uu^\top}{\beta}+\dfrac{B^2}{\sigma^2} \end{pmatrix}.

(D21.35)

All entries are bounded by a constant times $x_1^2$ (the base kernel is bounded on $K$ ), so $\E[\tau\Sigma^{-1}\tau]$ is finite. Under $\mu$ , $x_1=S$ and $u=U$ are independent, $\E x_1^2=\beta(\beta+1)$ , and the coordinates $u_j$ are i.i.d. uniform on $[-1,1]$ , with

\E u_j^2=\tfrac13,\qquad \E u_j^4=\tfrac15,\qquad \E u_j^6=\tfrac17,\qquad \E\bigl[u_j^{2a+1}\,h(u_1,\dots,u_{j-1},u_{j+1},\dots,u_m)\,k(u_j^2)\bigr]=0

(D21.36)

for bounded $h,k$ and $a\ge0$ , by the symmetry $u_j\mapsto-u_j$ of the law.

Top-left entry. $\E x_1^2\bigl(\tfrac1\beta+\tfrac{m/3}{\sigma^2}\bigr)=(\beta+1)+\beta(\beta+1)\cdot\tfrac m3\cdot\tfrac3{\beta(\beta+1)} =\beta+1+m=\beta+n$ ; dividing by $\Sigma_{11}=\beta$ gives $1+n/\beta$ , in agreement with Theorem D21.3(ii).

Off-diagonal block. $\E u=0$ and $\E[u^\top B]=\E\bigl[\abs u^2u^\top+\tfrac\beta2u^\top D\bigr]$ has $j$ th entry $\E[\abs u^2u_j]+\tfrac\beta2\E[u_j(1-u_j^2)]=0$ , each term being odd in $u_j$ . So the block vanishes.

Lower block. $\E[uu^\top]=\tfrac13\Id_m$ . Expanding, $B^2=\abs u^2uu^\top+\tfrac\beta2\bigl(uu^\top D+Duu^\top\bigr)+\tfrac{\beta^2}4D^2$ , whose $(j,k)$ entry is

\abs u^2u_ju_k+\tfrac\beta2u_ju_k\bigl(2-u_j^2-u_k^2\bigr)+\tfrac{\beta^2}4(1-u_j^2)^2\delta_{jk}.

(D21.37)

For $j\ne k$ every term is odd in $u_j$ , so its expectation vanishes. For $j=k$ ,

\gamma:=\E\bigl[(B^2)_{jj}\bigr] =\E\bigl[\abs u^2u_j^2\bigr]+\beta\,\E\bigl[u_j^2(1-u_j^2)\bigr]+\tfrac{\beta^2}4\E\bigl[(1-u_j^2)^2\bigr] =\Bigl(\tfrac15+\tfrac{m-1}9\Bigr)+\tfrac{2\beta}{15}+\tfrac{2\beta^2}{15},

(D21.38)

using $\E[\abs u^2u_j^2]=\E u_j^4+\sum_{k\ne j}\E u_k^2\E u_j^2$ , $\E[u_j^2-u_j^4]=\tfrac13-\tfrac15$ and $\E[1-2u_j^2+u_j^4]=1-\tfrac23+\tfrac15=\tfrac8{15}$ . Hence $\E B^2=\gamma\Id_m$ with

9\gamma=\tfrac95+(m-1)+\tfrac{6\beta}5+\tfrac{6\beta^2}5 =\frac{6\beta^2+6\beta+5m+4}5=\frac{6\beta^2+6\beta+5n-1}5 .

(D21.39)

The lower block of $\E[\tau\Sigma^{-1}\tau]$ is therefore $\beta(\beta+1)\bigl[\tfrac1{3\beta}+\tfrac{\gamma}{\sigma^2}\bigr]\Id_m =\bigl[\tfrac{\beta+1}3+3\gamma\bigr]\Id_m$ , and after normalization by $\sigma^{-2}$ on both sides,

G'=\frac3{\beta(\beta+1)}\Bigl[\frac{\beta+1}3+3\gamma\Bigr] =\frac1\beta+\frac{9\gamma}{\beta(\beta+1)} =\frac{5(\beta+1)+6\beta^2+6\beta+5n-1}{5\beta(\beta+1)} =\frac{6\beta^2+11\beta+5n+4}{5\beta(\beta+1)} .

(D21.40)

Since $\Sigma^{-1/2}=\beta^{-1/2}\oplus\sigma^{-1}\Id_m$ and the off-diagonal block vanishes, this proves (D21.34).

The inequalities. Since $\beta>0$ , $1+n/\beta\le2\iff n/\beta\le1\iff n\le\beta$ , which holds by hypothesis, with equality if and only if $\beta=n$ . Next,

G'\le2 \iff 6\beta^2+11\beta+5n+4\le10\beta^2+10\beta \iff 4\beta^2-\beta-5n-4\ge0 .

(D21.41)

Using $n\le\beta$ and then $\beta\ge n\ge2$ ,

4\beta^2-\beta-5n-4\ \ge\ 4\beta^2-6\beta-4\ =\ 2(2\beta+1)(\beta-2)\ \ge\ 0,

(D21.42)

so $G'\le2$ . The first inequality is strict unless $n=\beta$ and the second is strict unless $\beta=2$ ; hence $G'=2$ if and only if $n=\beta=2$ . At $n=\beta=2$ both eigenvalues equal 2 and the matrix is $2\,\Id_2$ . Since $\Sigma^{-1/2}\E[\tau\Sigma^{-1}\tau]\Sigma^{-1/2}\preceq2\Id_n$ is equivalent to $\E[\tau\Sigma^{-1}\tau]\preceq2\Sigma$ , every cube cone satisfies (16.15).

The product structure at $n=\beta=2$ . Here $K=[-1,1]$ , $C_K=\{x_1>\abs{x_2}\}$ and $\rho(x)=\tfrac12e^{-x_1}\one_{C_K}$ by Lemma D21.2(a). The linear map $A(c_1,c_2)=(c_1+c_2,c_1-c_2)$ has $\abs{\det A}=2$ , maps $(0,\infty)^2$ onto $C_K$ , and satisfies $e^{-c_1-c_2}=e^{-x_1}$ ; so if $c_1,c_2$ are i.i.d. standard exponential, $A(c_1,c_2)$ has density $e^{-x_1}/\abs{\det A}=\rho(x)$ on $C_K$ , i.e. $A_\#(\mathrm{Exp}(1)^{\otimes2})=\mu_{K,2}$ , and since $A(1,1)=(2,0)=\beta e_1$ , $\bar\mu_{K,2}=A_\#\bigl((\mathrm{Exp}(1)-1)^{\otimes2}\bigr)$ . As $A=\sqrt2\,R$ with $R$ the orthogonal map with columns $(e_1\pm e_2)/\sqrt2$ (a reflection, $\det R=-1$ ), in the orthonormal frame $(e_1\pm e_2)/\sqrt2$ the coordinates of $\bar x$ are $\sqrt2(c_1-1)$ , $\sqrt2(c_2-1)$ : independent centered exponential laws. (Since the i.i.d. product law is invariant under the coordinate swap $\sigma(c_1,c_2)=(c_2,c_1)$ , one may also write $\bar\mu_{K,2}=(A\sigma)_\#\bigl((\mathrm{Exp}(1)-1)^{\otimes2}\bigr)$ with $A\sigma=\sqrt2\,R\sigma$ and $R\sigma$ a rotation.)

5. Remarks: hypotheses, conventions, and what is not claimed¶

Remark D21.2 (Hypotheses actually used)

(1) $\beta\ge n$ is used only for log-concavity (Lemma D21.2(b)) and for the inequalities $1+n/\beta\le2$ and $G'\le2$ ; every identity holds for all $\beta>0$ . (2) The barycenter condition on $K$ is used for $\E U=0$ (centering, the block-diagonal $\Sigma$ , the vanishing odd moments) and to place the uniform law on $K$ in the hypotheses of Theorem 4.1 and Theorem D21.1. (3) From Theorem 4.1 only the base facts of Lemma D21.3 are used; the Stein identity for the cone itself is proved here (Lemma D21.4), since the cone is not a compact target. (4) Theorem D21.1 (uniqueness up to translation among essentially-continuous potentials, the class in which “moment potential” is defined in §0. Standing conventions) is used to identify the constructed potentials — $\varphi$ , the rescaled base potential, the product potential of the cube, and $\phi_T$ , all finite and therefore in that class — with the moment potential, and thereby the constructed kernels with the canonical ones. Without it, every statement remains true with “the moment potential” read as “the moment potential $\varphi$ of (D21.11)”. The restriction to essentially-continuous potentials is not cosmetic: uniqueness fails among all convex $\psi$ with $\int e^{-\psi}=1$ and $(\nabla\psi)_\#(e^{-\psi}\dd y)=\eta$ . For $\eta=\mathrm{Unif}[-1,1]$ and $c>1$ put $T_c=\tfrac2{\sqrt c}\operatorname{artanh}(1/\sqrt c)$ and $\psi_c(y)=\log\tfrac4c+2\log\cosh(\tfrac{\sqrt c}2y)$ on $[-T_c,T_c]$ , $\psi_c=+\infty$ outside. Then $\psi_c$ is convex and lower semi-continuous, $\psi_c'=\sqrt c\tanh(\tfrac{\sqrt c}2y)$ increases from -1 to 1 on $[-T_c,T_c]$ , $\psi_c''=\tfrac c2\cosh^{-2}(\tfrac{\sqrt c}2y)=2e^{-\psi_c}$ , so $\int e^{-\psi_c}=\tfrac12[\psi_c']_{-T_c}^{T_c}=1$ and the pushforward of $e^{-\psi_c}\dd y$ under $\psi_c'$ has density $e^{-\psi_c}/\psi_c''=\tfrac12$ on $(-1,1)$ ; but $\psi_c$ is discontinuous at $\pm T_c$ (finite there, $+\infty$ beyond), hence not essentially continuous, and it is not a translate of the finite potential of Lemma D21.7.

Remark D21.3 (Conventions on domains)

The closed forms $\calE_\Sigma$ , $\calE_\tau$ and their operators $\Aop_1$ , $\Aop$ are the closures from the core $\mathscr C=\R+C_c^\infty(\R^n)$ restricted to the open cone, which is the no-flux (Neumann-type) reading of §Conventions, domains, and affine covariance, the same core used in the proof of Theorem 16.1, and the one under which the Stein identity holds for test functions not vanishing on $\partial C_K$ . Under this convention Lemma D21.5 proves $x_1\in\Dom(\Aop)$ , so the phrase “the CMH Rayleigh quotient at the linear test function $g=x_1$ ” in Proposition 17.2(ii) is a statement about an admissible test function of Definition 16.1, not merely a formal evaluation. If the closed form were instead generated from $C_c^\infty(C_K-\beta e_1)$ (a Dirichlet-type reading), the value of the quotient at $g=x_1$ would be unchanged but the membership $g\in\Dom(\Aop)$ would have to be re-examined; the dossier does not address that reading.

Remark D21.4 (What is proved about the Hodge decomposition)

Theorem D21.3(i) verifies the hypotheses of Proposition 16.2 for $g=x_1$ — $g\in\Dom(\Aop)$ and $u=\tau\nabla g=x\in L^2(\mu;\Sigma^{-1})$ since $\E x^\top\Sigma^{-1}x=\beta+n<\infty$ — and computes the objects of that proposition directly: $h=\Aop g=\bar x_1$ , $\Aop_1^{-1}h=\psi-\E\psi$ , $w=0$ . The proposition itself is not used; the identity $\E\inner u{\Sigma^{-1}u}=\E\inner{\Sigma\nabla\psi}{\nabla\psi}$ it would yield is here the trivial consequence of $u=\Sigma\nabla\psi$ .

Remark D21.5 (What is proved about the third-moment decomposition)

Theorem D21.3(iii) proves the pointwise identity $\tau_Ze_1=e_1+\tfrac12T_3(e_1)Z$ directly. Lemma 16.2 is not invoked and its hypothesis $\E_\nu\norm{D^2\varphi}_{\HS}^2<\infty$ is not verified here for a general base $K$ (it is immediate for the cube, where $\tau_K$ is bounded, by Lemma D21.4(a)). The statement “ $v_{e_1}=0$ ” is understood as: the residual $\tau_Ze_1-e_1-\tfrac12T_3(e_1)Z$ , which is the quantity that lemma calls $v_{e_1}$ , vanishes identically. Likewise, for a general base only the $(1,1)$ entry of $\E[\tau\Sigma^{-1}\tau]$ is shown finite; finiteness of the whole matrix requires $\E\norm{\tau_K(U)}_{\HS}^2<\infty$ and is not needed for Proposition 17.2.

Obstructions respected. None of the three nodes carries a bounded_by fence. The statements are exact computations on a specific family and assert no universal bound; in particular they do not claim (16.15) beyond the cube cones and the axis direction of a general cone, and they say nothing about $\CMH$ itself.

References¶

Cordero-Erausquin, D., & Klartag, B. (2015). Moment Measures. Journal of Functional Analysis, 268(12), 3834–3866. 10.1016/j.jfa.2015.04.001
Klartag, B. (2014). Logarithmically-Concave Moment Measures I. In Geometric Aspects of Functional Analysis (Vol. 2116, pp. 231–260). Springer. 10.1007/978-3-319-09477-9_16