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Logarithmically Concave Distribution

A full-dimensional logarithmically concave distribution on R^n is a statistical distribution whose probability density function can be written as e^(-V(x)), where V:R^n->R union {infty} is a convex function. Allowing V=infty permits the density to vanish outside a convex set. Uniform distributions on convex bodies are examples.

Such a distribution is isotropic when its mean is zero and its covariance matrix is the identity matrix. The Kannan-Lovász-Simonovits conjecture asserts a universal bound on its Poincaré constant in this normalization (Bizeul et al. 2026).

For a real random variable X with a density that is a logarithmically concave function, mean zero, and variance 1, Pinelis (2026) proved the sharp bound P(X>=1)<=(1-1/sqrt(3))/2. Equality is attained by the uniform distribution on [-sqrt(3),sqrt(3)]. Pinelis (2026) credits ChatGPT with Lemmas 20 and 21 and a streamlined proof of Proposition 18.

See also

Convex Body, Differential Entropy, Kannan-Lovász-Simonovits Conjecture, Logarithmic Sobolev Inequality, Logarithmically Concave Function, Logarithmically Concave Sampling, Poincaré Constant, Statistical Distribution, Subgaussian Random Variable

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References

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.Pinelis, I. "Exact Bounds on the Distribution Function of Isotropic Log-Concave Distributions." 7 Oct 2026. https://arxiv.org/abs/2610.10289.

Cite this as:

Weisstein, Eric W. "Logarithmically Concave Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LogarithmicallyConcaveDistribution.html

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