KLS after its proofs

With the three proofs — of Bizeul–Klartag–Lehec (BKL, Chapter Bizeul–Klartag–Lehec: cumulants and suspension), of the second version of Song–Zhang (SZ v2, Chapter Song–Zhang, second version: repeated refinement with summable losses) and of Balasubramanian–Kasiviswanathan (BK, Chapter Balasubramanian–Kasiviswanathan: compatible integration) — the statement Conjecture 0.1 holds: there is a universal $C$ with $\CP(\mu)\le C$ for every isotropic log-concave probability measure $\mu$, in every dimension. This chapter collects what that statement gives, which constants the three proofs provide, and what is decided about the best constant. It is a guide to the theorem as a tool.

Equivalent forms, and the affine change of variables¶

The isotropic, affine, Cheeger and geometric forms of the theorem are stated in Section The conjecture, with the two-sided comparison (0.3) between the Poincaré and Cheeger constants; the geometric form is the one in which Kannan, Lovász and Simonovits asked the question Kannan et al., 1995. One passage is worth writing out, because it produces a stronger inequality on the way. The affine form (0.7) follows from the isotropic one by a change of variables. If $\mu$ is full-dimensional with covariance $\Sigma$, write $X=m+\Sigma^{1/2}Y$ with $Y$ isotropic and log-concave; for $f$ on $\R^n$ put $g(y)=f(m+\Sigma^{1/2}y)$. Then $\Var f(X)=\Var g(Y)\le C\,\E\abs{\nabla g(Y)}^2=C\,\E\abs{\Sigma^{1/2}\nabla f(X)}^2\le C\norm\Sigma_\op\E\abs{\nabla f(X)}^2$. A measure supported on a proper affine subspace is treated in that subspace. The middle term is the stronger, affine-invariant form $\Var_\mu f\le C\,\E_\mu\inner{\Sigma\nabla f}{\nabla f}$, whose best constant is the affine Poincaré constant $\CPaff(\mu)$ used in Chapter The moment map: the deterministic inequality.

What follows from it¶

Thin shell and slicing. Applied to $f(x)=\abs x^2$ in isotropic position, the theorem gives $\Var\abs X^2\le4n\,\CP(\mu)\le4Cn$ ((0.8)), the thin-shell bound; thin shell in turn implies a universal bound on the isotropic constant, the slicing problem Eldan & Klartag, 2011. This is the chain (0.11). Both consequences were proved before KLS, by other means (Section Solved neighbors that are not KLS): thin shell by Klartag and Lehec Klartag & Lehec, 2025, with the sharp constant $8n$, attained by products of exponentials, in a preprint of Chen and Klartag Chen & Klartag, 2026; slicing by Klartag and Lehec Klartag & Lehec, 2025, and again by Bizeul Bizeul, 2025. KLS now gives them a common source, with a constant $4C$ in the thin-shell bound that is not sharp.

Concentration. A Poincaré inequality with constant $\CP$ implies exponential concentration for Lipschitz functions: for $f$ 1-Lipschitz, $\mu(\abs{f-\E_\mu f}\ge t)\le2e^{-ct/\sqrt{\CP}}$ with $c$ universal Bobkov & Ledoux, 1997Bakry et al., 2014. With KLS, every 1-Lipschitz function of an isotropic log-concave vector deviates from its mean by more than $t$ with probability at most $2e^{-c't}$, with $c'$ independent of the dimension. For the Euclidean norm this is the thin-shell statement again, at the level of tails.

Mixing of the random walks on convex bodies. The algorithms that sample a convex body or estimate its volume run random walks — the ball walk, hit-and-run — whose mixing is controlled through conductance, and the conductance of these walks is bounded below in terms of the isoperimetric constant of the body. A dimension-free Cheeger constant in isotropic position therefore removes the corresponding factor from those mixing bounds. The arguments and their quantitative forms are surveyed in Lee & Vempala, 2024Klartag & Lehec, 2025.

The constants the three proofs give¶

The constant each proof gives is recorded in the row Constant obtained of the table of Section The three proofs side by side. For the question of the best constant, one fact matters: the BK bound is fully numerical, $\CP\le1+2\cdot10^{16}$ for every isotropic log-concave measure (Theorem 11.2), with inverse Cheeger scale at most $\sqrt{\pi(1+2\cdot10^{16})}$ (Corollary 11.3), while the BKL and SZ v2 arguments give universal constants that are not evaluated.

The question of the constant¶

Write $C_\star$ for the best universal constant: the supremum of $\CP(\mu)/\norm{\Cov\mu}_\op$ over all log-concave $\mu$ in all dimensions, equivalently the supremum of $\CP(\mu)$ over isotropic log-concave $\mu$. The three proofs give $C_\star<\infty$, and BK gives $C_\star\le1+2\cdot10^{16}$. Here is what is decided about its value, and where.

Testing linear functions gives $C_\star\ge1$, with equality for the Gaussian, and the exponential on the line gives $C_\star\ge4$ (Section Examples by hand). The value 4 is not exceeded on the line, on products of one-dimensional log-concave laws, or on log-concave Dirichlet laws and their linear images, products and convolutions (Sections Examples by hand and A deterministic inequality, sharp on products and Dirichlet laws). For the Dirichlet laws the bound is $\CPaff\le4$, and $\inner{\Sigma\nabla f}{\nabla f}\le\norm\Sigma_\op\abs{\nabla f}^2$ turns it into $\CP\le4\norm{\Cov}_\op$. By Theorem 16.1 and Proposition 15.1, a universal bound $\CMH\le4$ would give $\CP\le4\norm{\Cov}_\op$ for every log-concave measure, hence $C_\star=4$.

On the exponential the ratio 4 is reached only as a supremum, since no square-integrable function attains its Poincaré constant. Between 4 and $1+2\cdot10^{16}$ this manuscript decides nothing: it neither proves $C_\star=4$ nor exhibits a log-concave measure with ratio above 4, and none of the three proofs gives the upper bound 4. The moment-Hessian inequality is the one mechanism here aimed at the value 4. Its linear test — the inequality restricted to linear functions, $\E H^2\preceq4\,\Id$ (Section A deterministic inequality, sharp on products and Dirichlet laws), not the bound $\CP\ge\norm{\Cov}_\op$ above — has a sharp form with constant 2, Conjecture 16.2, which none of the three proofs gives.

What comes next¶

The theorem leaves the mechanisms by which it might be proved otherwise, and the further statements they aim at. Chapter Alternative mechanisms after KLS maps three of them by what each would add to the three proofs: the moment map, a deterministic proof with the constant 4; the fixed eigenfunction, a localization mechanism that ignores covariance spikes; and conditional fibers, a mechanism resting only on one-dimensional inequalities.