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Poincaré Constant

The Poincaré constant C_P(mu) of a probability measure on R^n is the least constant for which

Var_mu(f)<=C_P(mu)int||del f||^2dmu

for every sufficiently regular function f, where Var_mu(f)=intf^2dmu-(intfdmu)^2 is its variance. This is the Poincaré inequality for mu.

The Kannan-Lovász-Simonovits conjecture asserts a universal upper bound on C_P(mu) for isotropic logarithmically concave distributions. Bizeul et al. (2026) reported a proof of this bound.

See also

Kannan-Lovász-Simonovits Conjecture, Logarithmically Concave Distribution, Poincaré Inequality, Probability Measure

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References

Balasubramanian, K. and Kasiviswanathan, S. "A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures." 8 Oct 2026. https://arxiv.org/abs/2610.07728v2.Bizeul, P.; Klartag, B.; and Lehec, J. "Presenting a Proof of the Kannan-Lovász-Simonovits Conjecture." 4 Oct 2026. https://arxiv.org/abs/2610.05474.

Cite this as:

Weisstein, Eric W. "Poincaré Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoincareConstant.html

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