Exact constants on the line and on products, and the bound $4$ on Dirichlet laws

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 $\mathrm{CMH}(4)$ in three classes where the constant of Definition 16.1 can be computed: the line (Theorem 17.1), products (Theorem 17.2, Corollary 17.1) and the log-concave Dirichlet family (Theorem 17.3, with Lemma 17.1, Lemma 17.2, Lemma 17.3, Corollary 17.2, Corollary 17.3). It also records Corollary 17.5. The Dirichlet case is proved by lifting to independent Gamma variables, completing a square in the Gamma Bochner identity, and using the Euler constraint of degree-0 homogeneity. Universal $\mathrm{CMH}(4)$ remains open. Products of centered one-sided exponentials saturate the constant 4 exactly.

On the line, $\CMH=\CP/\Var$ exactly, the KLS bound gives $\CP\le4\Var$ , and the centered exponential attains 4 (Theorem D29.1).
For products, the cross terms $\E[(L_ig)(L_jg)]$ are nonnegative, so $\CMH$ is the maximum over the factors (Theorem D29.2). Affine Poincaré passes to linear images (Corollary D29.1).
The Dirichlet data and the tangent pseudo-inverse (Lemma D29.1) reduce the claim to $A(A+1)d_\alpha\le4n_\alpha$ .
Gamma lift: the Bochner identity (D29.14) and a square completion give $N_\Gamma-\tfrac14D_\Gamma=\sum_i\E R_i+\sum_i\delta_iD_i$ (Lemma D29.2). The Euler constraint bounds each $R_i$ from below (Lemma D29.3).
Using $S\perp P$ and the inverse moments of $S$ , step 4 becomes a pointwise coefficient $F_A(\alpha_i,P_i)$ . A scalar minimization bounds it by $A-\tfrac12+s_A$ with $s_A\ge0$ (Lemma D29.4), which proves Theorem D29.3, with strict surplus for $A>3$ .
Theorem 16.1 with step 2 gives $\CPaff\le4$ for Dirichlet laws, their products, linear images and convolutions.

Scope. This dossier proves $\mathrm{CMH}(4)$ in the three classes where the constant of Definition 16.1 is computable, the third being the first genuinely nonproduct family for which Route C has a theorem. Combined with Theorem 16.1 (dossier solutions/thm-cmh-normalization.md) each result yields an affine Poincaré bound with the same constant. Nothing here bears on universal $\mathrm{CMH}(4)$ , which remains open; Corollary 17.5 in the manuscript explains why the product case in particular is a warning rather than encouragement.

Notation is that of §The Stein generator and the CMH constant: $H$ is the moment-map Stein kernel in target coordinates, $L_\mu=\Div_\mu(H\nabla\,\cdot\,)$ , $\Aop=-L_\mu$ , and $\CMH$ is (16.5).

1. The line¶

Let $\mu(\dd x)=\rho\dd x$ be centered on $(\ell,r)$ with variance $\sigma^2$ and Stein kernel $\tau$ , the zero-flux solution of $(\tau\rho)'=-x\rho$ .

Proof D29.1

We use the natural no-flux realizations of §Conventions, domains, and affine covariance. Let $D$ be the closed maximal derivative in $L^2(\mu)$ , with domain $H^1(\mu)$ , and let $\Aop$ be the operator of the natural weighted form $\mathcal E_\tau(f,q)=\int\tau f'q'\,d\mu$ . Constants and bounded locally absolutely continuous functions whose derivative has compact interior support belong to these form domains when their displayed energies are finite. The local density regularity is that of the differential-operator setup (in particular, locally positive continuous densities suffice, using weak derivatives); no endpoint regularity, log-concavity, or spectral gap is used. Both constants in the asserted identity are allowed to be infinite, and the CMH numerator is understood as an extended nonnegative integral.

Put $p=\tau\rho$ . Centering and nonzero finite variance give $p>0$ in the interior. The following graph core encodes the adjoint’s no-flux condition:

\mathscr U=\{u=\phi/\rho:\phi\in C_c^\infty((\ell,r))\}, \qquad D_\mu^*u=-\phi'/\rho.

(D29.1)

To verify the core assertion, define $B_0$ by this expression on $\mathscr U$ . Its domain is dense: compact localization and smooth approximation of the flux $\rho u$ are dense in $L^2(\rho^{-1}dx)$ . The adjoint condition, tested against $\phi$ , is precisely that a scalar function have a distributional derivative in $L^2(\mu)$ . Thus $B_0^*=D$ , with no boundary restriction on the scalar function, and $\overline{B_0}=D_\mu^*$ . Also $\ker D$ is exactly the constants.

Every field of this core is attained by an actual CMH test. Fix an interior $x_0$ and set

f_\phi(x)=\int_{x_0}^x\frac{\phi(t)}{p(t)}\,dt, \qquad h_\phi=-\frac{\phi'}\rho.

(D29.2)

Then $f_\phi$ is bounded and constant near the endpoints, and $\mathcal E_\tau(f_\phi,f_\phi)=\int\phi^2/p\,dx<\infty$ . For every weighted-form test $q$ , compactly supported integration by parts gives

\mathcal E_\tau(f_\phi,q)=\int\phi q'\,dx =-\int\phi' q\,dx=\inner{h_\phi}{q}.

(D29.3)

Since $h_\phi\in L^2(\mu)$ , the operator-domain criterion gives $f_\phi\in\Dom(\Aop)$ , $\Aop f_\phi=h_\phi$ , and $\tau f_\phi'=u$ . Moreover $h_\phi$ is centered. No replacement of $\operatorname{Ran}\Aop$ by all of centered $L^2$ has been made.

Lower bound, including the infinite case. Put $M=\sigma^2\CMH(\mu)$ . If $M<\infty$ , these attained tests give $\norm u_2^2\le M\norm{D_\mu^*u}_2^2$ on $\mathscr U$ , and graph-core closure extends it to all of $\Dom(D_\mu^*)$ . This bound and closedness make $\operatorname{Ran}D_\mu^*$ closed: if $D_\mu^*u_j$ converges, then $u_j$ is Cauchy and the closed graph supplies its limiting preimage. Since the closure of this range is $(\ker D)^\perp=L^2_0(\mu)$ , the range equals $L^2_0(\mu)$ . For any centered $v\in\Dom D$ , choose $u$ with $D_\mu^*u=v$ . Then

\norm v_2^2=\inner{Dv}{u} \le\sqrt M\norm{Dv}_2\norm v_2.

(D29.4)

Thus $\CP\le M$ . If $M=\infty$ that inequality is automatic; in particular $\CP=\infty$ forces $\CMH=\infty$ .

Upper bound when $\CP<\infty$ . Given $f\in\Dom(\Aop)$ , put $h=\Aop f$ . Constants belong to $\ker\Aop$ , so $h$ is centered. Poincaré gives, for every $q\in\Dom D$ ,

|\inner hq|\le\sqrt{\CP}\norm h_2\norm{Dq}_2.

(D29.5)

Apply Riesz representation to the bounded functional $Dq\mapsto\inner hq$ on the closure of $\operatorname{Ran}D$ . There exists $w\in L^2(\mu)$ such that

\inner w{Dq}=\inner hq\quad(q\in\Dom D), \qquad \norm w_2\le\sqrt{\CP}\norm h_2.

(D29.6)

For $\eta\in C_c^\infty((\ell,r))$ , its bounded primitive $q(x)=\int_{x_0}^x\eta(t)\,dt$ belongs to both scalar form domains. The defining weak identity for $\Aop f=h$ therefore yields

\int\tau f'\eta\,d\mu=\inner hq=\int w\eta\,d\mu.

(D29.7)

The left integrand is locally integrable by weighted Cauchy--Schwarz. Arbitrariness of $\eta$ proves $\tau f'=w$ almost everywhere, without assuming square integrability of $\tau f'$ in advance. Consequently $\norm{\tau f'}_2^2\le\CP\norm{\Aop f}_2^2$ for every CMH test. Since $H=\tau$ and $\Sigma=\sigma^2$ , this proves $\sigma^2\CMH\le\CP$ in the finite case. In the infinite case equality was already forced by the lower bound. No operator inverse is used.

The one-dimensional specialization of the Kannan–Lovász–Simonovits bound Kannan et al., 1995, recorded for example in Cattiaux & Guillin, 2018, Eq. (2.25) with that attribution, is $\CP(\mu)\le4\Var(\mu)$ .

For sharpness let $X=Y-1$ with $Y\sim\mathrm{Exp}(1)$ . Then $X$ is centered, has density $e^{-(x+1)}\one_{x\ge-1}\dd x$ , and $\Var(X)=1$ . For $0<a<\tfrac12$ set $f_a(x)=e^{a(x+1)}$ . Direct integration gives

\frac{\Var(f_a(X))}{\E\abs{f_a'(X)}^2} =\frac{(1-2a)^{-1}-(1-a)^{-2}}{a^2(1-2a)^{-1}} =\frac1{(1-a)^2}\longrightarrow4.

(D29.8)

The upper bound therefore gives $\CP(X)=4$ , and the identity already proved gives $\CMH(X)=4$ .

2. Products¶

Proof D29.2

Here $\Sigma=\diag(\Sigma_1,\dots,\Sigma_m)$ , $H=\diag(H_1,\dots,H_m)$ , and $L_\mu=\sum_iL_i$ with $L_i$ acting in block $i$ only, so the $L_i$ are commuting nonpositive generators. For $i\ne j$ , two integrations by parts give

\E[(L_ig)(L_jg)] =\E\Tr\!\left(H_iD^2_{ij}g\,H_jD^2_{ji}g\right) =\E\norm{H_i^{1/2}D^2_{ij}g\,H_j^{1/2}}_{\HS}^2\ge0.

(D29.10)

Hence $\E(L_\mu g)^2=\sum_i\E(L_ig)^2+2\sum_{i<j}\E[(L_ig)(L_jg)]\ge\sum_i\E(L_ig)^2$ .

For the numerator, conditioning on the other blocks and using the defining inequality for $\CMH(\mu_i)$ gives

\E\inner{H_i\nabla_i g}{\Sigma_i^{-1}H_i\nabla_i g} \le\CMH(\mu_i)\,\E(L_i g)^2.

(D29.11)

Summing and using the previous display gives “ $\le$ ”. Testing on $g$ depending on one block gives “ $\ge$ ”. The one-dimensional formula, including factors with infinite Poincaré or CMH constant, follows from the extended-constant identity Theorem D29.1; the affine-image consequence uses the invertible affine covariance of $\CMH$ from §Conventions, domains, and affine covariance.

3. The Dirichlet family: setup¶

Let $P\sim\Dir(\alpha)$ , $\alpha_i\ge1$ , $A=\sum_i\alpha_i$ , $q=\alpha/A$ . With $C(p)=\diag(p)-pp^\top$ and $L_\alpha$ the Wright–Fisher generator (17.9), the potential $\varphi(y)=A\log\sum_ie^{y_i/A}-q\cdot y$ has $\nabla\varphi=p-q$ and $D^2\varphi=C(p)/A$ , and pushes $e^{-\varphi}$ forward to the centered law $P-q$ . Hence the canonical data are $H=C(p)/A$ , $L_\mu=L_\alpha/A$ , and $\Sigma$ as in (17.10).

With $u_i=(C(P)\nabla g)_i$ , $d_\alpha$ , $n_\alpha$ as in (17.12), the statement $\CMH\le4$ becomes exactly $A(A+1)d_\alpha(g)\le4n_\alpha(g)$ : the numerator is $A^{-2}\cdot A(A+1)d_\alpha=\tfrac{A+1}Ad_\alpha$ and the denominator $A^{-2}n_\alpha$ .

4. The Gamma lift¶

Let $Y_i\sim\GammaLaw(\alpha_i,1)$ independent, $S=\sum_iY_i$ , $P=Y/S$ ; then $S\sim\GammaLaw(A,1)$ , $P\sim\Dir(\alpha)$ , and $S\perp P$ . Set $G(Y)=g(Y/S)$ , homogeneous of degree 0, and let $\calL_\Gamma G=\sum_i(Y_iG_{ii}+(\alpha_i-Y_i)G_i)$ be the Laguerre generator. To identify the source of its integrated Bochner identity, center the product-Gamma law by writing $X_i=Y_i-\alpha_i$ . Its canonical moment Hessian, expressed in the unchanged $Y$ coordinates, is

H_\Gamma=\diag(Y_1,\ldots,Y_m).

(D29.12)

Indeed, writing $\rho_\Gamma$ for the product-Gamma density,

\Div_\Gamma(H_\Gamma\nabla G) =\sum_i\rho_\Gamma^{-1}\partial_i(\rho_\Gamma Y_iG_i) =\sum_i\bigl(Y_iG_{ii}+(\alpha_i-Y_i)G_i\bigr)=\calL_\Gamma G.

(D29.13)

Thus Proposition 16.1, specialized to $H_\Gamma$ , gives

N_\Gamma(G):=\E(\calL_\Gamma G)^2 =\E\left[\sum_iY_iG_i^2+\sum_{i,j}Y_iY_jG_{ij}^2\right].

(D29.14)

Here the second sum is over ordered pairs: because $D^2G$ is symmetric, each off-diagonal square occurs twice, exactly as in $\Tr(H_\Gamma D^2G\,H_\Gamma D^2G)$ . There is no additional factor of 2 or $1/2$ . Finally set $D_i(G)=\E[Y_i^2G_i^2]/\alpha_i$ and $D_\Gamma=\sum_iD_i$ .

Proof D29.5

The occurrence of the first-order row $\E[Y_iG_i^2]$ and the full ordered Hessian row $\E[\sum_jY_iY_jG_{ij}^2]$ below is precisely the $i$ th-row contribution to the specialized Bochner identity (D29.14). For $Y\sim\GammaLaw(a,1)$ and smooth $u$ with the usual decay,

\E[Y^2uu']=\tfrac12\E[Y^2(u^2)'] =\tfrac1{2\Gamma(a)}\int y^{a+1}e^{-y}(u^2)'\dd y =-\tfrac1{2\Gamma(a)}\int u^2\bigl((a+1)y^a-y^{a+1}\bigr)e^{-y}\dd y,

(D29.15)

i.e. $\E[Y^2uu']=-\tfrac12\E[((a+1)Y-Y^2)u^2]$ . Expanding the first square of $R_i$ and applying this with $u=G_i$ , $u'=G_{ii}$ , $a=\alpha_i$ , the cross term is

-\tfrac2{\alpha_i+1}\E[Y_i^2G_iG_{ii}] =\tfrac1{\alpha_i+1}\E[((\alpha_i+1)Y_i-Y_i^2)G_i^2] =\E[Y_iG_i^2]-\tfrac1{\alpha_i+1}\E[Y_i^2G_i^2].

(D29.16)

Therefore

\E R_i=\E\Bigl[Y_iG_i^2+\sum_jY_iY_jG_{ij}^2\Bigr] +\Bigl(\tfrac1{(\alpha_i+1)^2}-\tfrac1{\alpha_i+1}\Bigr)\E[Y_i^2G_i^2],

(D29.17)

and $\tfrac1{(\alpha_i+1)^2}-\tfrac1{\alpha_i+1}=-\tfrac{\alpha_i}{(\alpha_i+1)^2}$ . Summing over $i$ , the bracket sums to $N_\Gamma(G)$ and the correction to $-\sum_i\tfrac{\alpha_i^2}{(\alpha_i+1)^2}D_i(G)$ . Subtracting $\tfrac14D_\Gamma$ gives the claim, and $\delta_i=\tfrac{\alpha_i^2}{(\alpha_i+1)^2}-\tfrac14\ge0$ iff $\tfrac{\alpha_i}{\alpha_i+1}\ge\tfrac12$ iff $\alpha_i\ge1$ .

This lemma alone proves $\mathrm{CMH}(4)$ for the product-Gamma law; for that consequence, $\alpha_i\ge1$ is used only to make $\delta_i\ge0$ . In the Dirichlet proof below the exact $\delta_i$ term is instead retained inside $F_A(\alpha_i,P_i)$ , and log-concavity is consumed when Lemma D29.4 is applied with $a=\alpha_i\ge1$ .

Proof D29.6

Homogeneity of degree 0 gives Euler’s identity $\sum_jY_jG_j=0$ ; differentiating in $Y_i$ yields $G_i+\sum_jY_jG_{ji}=0$ , i.e. the single linear constraint $\sum_jY_jG_{ij}=-G_i$ . Set $t_i=Y_i(G_{ii}-G_i/(\alpha_i+1))$ and $t_j=Y_jG_{ij}$ for $j\ne i$ . Then

\sum_jt_j=\sum_jY_jG_{ij}-\tfrac{Y_i}{\alpha_i+1}G_i=-G_i\Bigl(1+\tfrac{Y_i}{\alpha_i+1}\Bigr), \qquad R_i=t_i^2+\sum_{j\ne i}\tfrac{Y_i}{Y_j}t_j^2 .

(D29.18)

The weights are $w_i=1$ and $w_j=Y_i/Y_j$ , so $\sum_jw_j^{-1}=1+\sum_{j\ne i}Y_j/Y_i=S/Y_i$ . Minimizing $\sum_jw_jt_j^2$ subject to $\sum_jt_j=c$ gives $c^2/\sum_jw_j^{-1}$ , which is the stated bound.

Homogeneity further gives $\calL_\Gamma G=S^{-1}L_\alpha g$ and $Y_iG_i=u_i(P)$ : indeed $G_i=\partial_{Y_i}g(Y/S)=S^{-1}(g_i-\sum_kP_kg_k)$ , so $Y_iG_i=P_ig_i-P_i\sum_kP_kg_k=u_i(P)$ . For $A>2$ , $\E S^{-1}=(A-1)^{-1}$ and $\E S^{-2}=z_A^{-1}$ with $z_A=(A-1)(A-2)$ . Since $S\perp P$ ,

N_\Gamma(G)=\E[S^{-2}]\,\E(L_\alpha g)^2=\frac{n_\alpha(g)}{z_A}, \qquad D_\Gamma(G)=\E\sum_i\frac{(Y_iG_i)^2}{\alpha_i}=d_\alpha(g),

(D29.19)

the second because $Y_iG_i$ depends on $P$ alone. Substituting $Y_i=SP_i$ and $G_i=u_i/(SP_i)$ into Lemma D29.3 and taking expectations gives (17.20).

5. The scalar minimization¶

Proof D29.7

The $p$ -dependent part of $F_A$ is $\phi_a(p)=a\bigl(p^{-1}+zp(a+1)^{-2}\bigr)$ , convex on $(0,\infty)$ with unconstrained minimizer $p_*=(a+1)/\sqrt z$ . Hence on $(0,1]$ the minimum is at $p=1$ when $p_*\ge1$ , i.e. $\sqrt z\le a+1$ , and at $p_*$ otherwise.

Case $z\le4$ . Then $\sqrt z\le2\le a+1$ for every $a\ge1$ , so $p=1$ and

F_A(a,1)=a+\frac{2a(A-2)}{a+1}+\frac{za}{a+1}-\frac z4 .

(D29.20)

Each of the three terms is increasing in $a$ on $[1,\infty)$ , so the minimum is at $a=1$ : $1+(A-2)+z/2-z/4=A-1+z/4=A-\tfrac12+\tfrac{z-2}4$ .

Case $z\ge4$ . At $p=p_*$ the two terms of $\phi_a$ are equal, so $\phi_a(p_*)=2a\sqrt z/(a+1)$ . Writing $r=a/(a+1)\in[\tfrac12,1)$ for $a\ge1$ ,

F_A=2r\sqrt z+2r(A-2)+zr^2-\frac z4=2r(\sqrt z+A-2)+zr^2-\frac z4,

(D29.21)

increasing in $r$ on $[0,1)$ , so the minimum is at $r=\tfrac12$ ( $a=1$ ) and equals $(\sqrt z+A-2)+z/4-z/4=A-2+\sqrt z=A-\tfrac12+\sqrt z-\tfrac32$ . In the complementary region $\sqrt z\le a+1$ the value $F_A(a,1)$ is increasing in $a$ and at the interface is not smaller, so the stated minimum stands.

Finally $A\ge3$ gives $z_A=(A-1)(A-2)\ge2$ , so $s_A=(z-2)/4\ge0$ when $z\le4$ and $s_A=\sqrt z-\tfrac32\ge2-\tfrac32>0$ when $z\ge4$ .

6. Proof of the Dirichlet theorem¶

Proof D29.8 (Proof of Theorem D29.3)

If $m=2$ and $A<3$ , the law is one dimensional and Theorem D29.1 applies (a $\mathrm{Beta}$ law is log-concave precisely when both parameters are $\ge1$ ). In every remaining case $A=\sum_i\alpha_i\ge3$ — automatically so when $m\ge3$ — hence $z_A\ge2>0$ and the inverse moments $\E S^{-1},\E S^{-2}$ are finite.

By Lemma D29.2 and (17.20), since $D_i(G)=\E[u_i^2]/\alpha_i$ , the coefficient of $\E_P[u_i^2]$ in $N_\Gamma(G)-\tfrac14D_\Gamma(G)$ is at least

\frac1{z_AP_i}+\frac2{(A-1)(\alpha_i+1)}+\frac{P_i}{(\alpha_i+1)^2}+\frac{\delta_i}{\alpha_i}.

(D29.22)

Multiply by $z_A\alpha_i$ and use $z_A/(A-1)=A-2$ together with $z_A\delta_i=z_A\alpha_i^2/(\alpha_i+1)^2-z_A/4$ :

\frac{\alpha_i}{P_i}+\frac{2\alpha_i(A-2)}{\alpha_i+1} +\frac{z_A\alpha_i(\alpha_i+P_i)}{(\alpha_i+1)^2}-\frac{z_A}4 =F_A(\alpha_i,P_i).

(D29.23)

Lemma D29.4 bounds this below by $A-\tfrac12+s_A$ pointwise in $P_i$ , so

N_\Gamma(G)-\tfrac14D_\Gamma(G) \ \ge\ \frac{A-\tfrac12+s_A}{z_A}\sum_i\frac{\E_P[u_i^2]}{\alpha_i} =\frac{A-\tfrac12+s_A}{z_A}\,d_\alpha(g).

(D29.24)

Substituting $N_\Gamma=n_\alpha/z_A$ and $D_\Gamma=d_\alpha$ , multiplying by $z_A$ , and using

\frac{z_A}4+A-\frac12=\frac{A^2-3A+2}4+\frac{4A-2}4=\frac{A(A+1)}4,

(D29.25)

we obtain $n_\alpha(g)\ge\bigl(\tfrac{A(A+1)}4+s_A\bigr)d_\alpha(g)$ . Since $s_A\ge0$ this gives $A(A+1)d_\alpha(g)\le4n_\alpha(g)$ .

Qualitative dimension-free KLS bounds for simplices and conservative Gamma models are prior art; see Kolesnikov & Milman, 2016, §1.2 as a secondary pointer. The specific content of this proof is the constant 4, the quantitative surplus, and the CMH-level estimate; no broader priority claim is made.

Closure. The argument is written for smooth $g$ with controlled boundary behavior. To close: take polynomials on the simplex, lift them after a radial cutoff $\{S\ge\eps\}$ so that all Gamma integrations by parts in Lemma D29.2 are justified, and let $\eps\downarrow0$ ; the inverse moments $\E S^{-1},\E S^{-2}$ used in (17.19) are finite because $A\ge3$ throughout the Gamma branch. Polynomials form a core for $L_\alpha$ , so the closed forms extend the inequality to $\Dom(L_\alpha)$ . Only the residual branch $m=2$ , $A<3$ uses the one-dimensional no-flux closure of Theorem D29.1 instead.

Obstructions respected. No bounded_by edge applies: the obstruction statements of the manuscript are scoped by their own statements to the fixed-cut Eldan program. The only one with method-level reach, rem:projection-ceiling, forbids deriving quadratic-chaos thin shell from radial or projection information alone; the Dirichlet proof uses the full Hessian row through the Euler constraint of Lemma D29.3, not projection tests, and makes no thin-shell claim. Consistency with the sharp external inputs holds: by Theorem 17.1, the boundary mechanism forcing the constant 4 is the one-dimensional exponential, which is also the extremal case of Theorem 4.2.

What this does not show. Theorem D29.3 is a family result, not evidence for universal $\mathrm{CMH}(4)$ . By Corollary 17.2 the Dirichlet family is strictly inside the bound, and by Theorem D29.2 the saturating cases are products of centered one-sided exponentials, where the slack is exactly zero. Whether that endpoint is perturbatively unstable is precisely the open issue just stated.

References¶

Kannan, R., Lovász, L., & Simonovits, M. (1995). Isoperimetric Problems for Convex Bodies and a Localization Lemma. Discrete & Computational Geometry, 13(3–4), 541–559. 10.1007/BF02574061
Cattiaux, P., & Guillin, A. (2018). On the Poincaré Constant of Log-Concave Measures.
Kolesnikov, A. V., & Milman, E. (2016). The KLS Isoperimetric Conjecture for Generalized Orlicz Balls.