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KLS Poincaré bounds for isotropic log-concave measures

For an isotropic log-concave random vector X and symmetric M, the quadratic-form KLS target is Var⟨MX,X⟩≤2 E|∇⟨MX,X⟩|². The broader question bounds the Poincaré constant for all observables; the stated general upper order is (log n)^(1/4).

Why it matters and what counts as progress

Why it matters: It separates a sharp variance inequality for quadratic forms from the full KLS problem for arbitrary observables.

What counts as progress: Prove the sharp quadratic-form variance bound or improve the dimension dependence of the full KLS bound; the quadratic case alone does not resolve full KLS.

How the problem developed

  1. 1995 · Conjecture [1]

    Ravindran Kannan, László Lovász and Miklós Simonovits Introduced the dimension-independent isoperimetric conjecture for convex bodies and log-concave measures, which has an equivalent Poincaré formulation.

  2. 27 Jul 2026 · Partial result [2]

    Brayden Letwin Proved a sharp quadratic-form variance inequality and derived an O(log^(1/4)n) upper bound for the KLS constant.

  3. 4 Oct 2026 · Proof claim [4][3]

    Pierre Bizeul, Boaz Klartag and Joseph Lehec Posted a proof of the full dimension-free KLS conjecture, using a recent criterion of Song and Zhang as a key step.

AI contributions 2

Sources

  1. Kannan, Lovász and Simonovits, Isoperimetric problems for convex bodies and a localization lemma
  2. Brayden Letwin, The KLS constant is O(log^(1/4)n)
  3. Zhao Song and Xinzhi Zhang, An O(4^(log*n)) Bound for the KLS Constant
  4. Pierre Bizeul, Boaz Klartag and Joseph Lehec, Presenting a proof of the KLS conjecture
  5. Bizeul–Klartag–Lehec: theorem and AI attribution
  6. Song–Zhang: criterion and concurrent claim
  7. Letwin’s quadratic-form KLS paper
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