KLS Conjecture for Quadratic Forms
Prior state unknown→proved
with constant 2; also improves the global KLS bound to $O(\log^{1/4} n)$
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geometry-topology / Asymptotic convex geometry
KLS Conjecture for Quadratic Forms
Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is $\operatorname{Var}\langle MX, X\rangle \le C\, \mathbb{E}|\nabla\langle MX, X\rangle|^2$ for every symmetric $M$?
45Significance / 100
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Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is $\operatorname{Var}\langle MX, X\rangle \le C\, \mathbb{E}|\nabla\langle MX, X\rangle|^2$ for every symmetric $M$?
with constant 2; also improves the global KLS bound to $O(\log^{1/4} n)$
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