Family 1: classical needle localization

KLS is now proved, by three different arguments compared in Chapter The proofs of KLS compared. This chapter describes what needle localization controls on its own and what it cannot give without them.

Object followed. One-dimensional restrictions of $\mu$ to segments — needles — carrying a log-concave weight.

What it buys. The KLS localization lemma reduces certain $n$-dimensional integral inequalities to inequalities on a segment Kannan et al., 1995. Schematically: given two integral constraints

it produces points $a,b\in\R^n$ and an affine $\ell>0$ such that both constraints remain valid for the one-dimensional measure

carried on the segment $x(t)=(1-t)a+tb$. The weight $\ell^{n-1}$ is log-concave, so the reduced problem is a one-dimensional log-concave problem, and one-dimensional log-concave isoperimetry is completely understood: for a log-concave $\nu$ on $\R$ with coordinate $T$,

Sharpest result. Combining (1.3) with localization gives

which in isotropic position ($\Tr\Cov\mu=n$) is the classical bound

the 1995 entry of the history table in Section The quantitative history.

What it does not reach alone. A covariance-sensitive decomposition, one whose needles inherit the operator covariance of $\mu$ and not merely one or two scalar integral constraints. The obstruction is a count. The localization construction preserves one or two scalar integral constraints, while isotropy is $n(n+1)/2$ constraints. Nothing forces an individual needle to be even approximately isotropic, and a single needle can have variance of order $n$ while the original measure has $\Cov\mu=I$. This is why (1.4) sees $\Tr\Cov\mu$, a sum of $n$ eigenvalues, where the affine form (0.7) asks for $\norm{\Cov\mu}_\op$, the largest one alone. The gap between $\Tr$ and $\norm{\cdot}_\op$ in (1.4) is the factor $\sqrt n$.

Klartag’s needle decomposition, built from transport rays rather than from the bisection construction, is substantially more geometric and can be arranged so that a chosen mean-zero function remains mean-zero on almost every needle Klartag, 2014. This strengthens the construction, since the preserved constraint is chosen adaptively rather than arbitrarily, but it does not deliver uniformly bounded conditional covariance on the needles.

Where this family meets the alternative mechanisms. Not directly: none of the three alternative mechanisms, nor the fixed cut of the archive, decomposes $\mu$ into one-dimensional pieces. The moment map (Chapter The moment map: the deterministic inequality) is the only deterministic mechanism here, but it works in the moment-map coordinates of Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels rather than by needle decomposition. What the family contributes is the diagnosis — global isotropy is not inherited needle by needle — which is the first instance of the fixed-versus-adaptive difficulty of Section Obstacles for alternative arguments.