Log-concave distribution
A probability distribution on (\mathbb{R}^n) is log-concave when its density, or more generally its probability measure, satisfies a multiplicative form of concavity under linear interpolation. Log-concave distributions connect convex analysis with probability theory, and their principal structural properties follow from the Prékopa–Leindler inequality. They include Gaussian measures, uniform measures on convex bodies, and numerous distributions arising from convex optimization and statistical mechanics.
Definition
A nonnegative function (f:\mathbb{R}^n\to[0,\infty)) is log-concave if
[ f\bigl((1-\lambda)x+\lambda y\bigr) \geq f(x)^{1-\lambda}f(y)^\lambda ]
for every (x,y\in\mathbb{R}^n) and every (\lambda\in[0,1]). The definition uses the standard extended-real conventions at points where (f) vanishes.
On the set where (f) is positive, this condition is equivalent to the concavity of (\log f). Equivalently, a log-concave function can be represented as
[ f(x)=e^{-V(x)}, ]
where (V:\mathbb{R}^n\to(-\infty,\infty]) is a convex function. Allowing (V) to take the value (+\infty) incorporates densities supported on proper convex subsets of (\mathbb{R}^n).
A probability distribution with a density (f) relative to Lebesgue measure is called log-concave when (f) is log-concave. The measure-theoretic definition does not require a full-dimensional density. A Borel measure (\mu) is log-concave if
[ \mu\bigl((1-\lambda)A+\lambda B\bigr) \geq \mu(A)^{1-\lambda}\mu(B)^\lambda ]
for nonempty compact sets (A,B\subseteq\mathbb{R}^n), where
[ (1-\lambda)A+\lambda B
{(1-\lambda)a+\lambda b:a\in A,\ b\in B}. ]
This formulation includes measures concentrated on affine subspaces. Subject to the usual identification with the affine hull of the support, a log-concave measure has a log-concave density with respect to Lebesgue measure on that affine hull.
Geometric structure
The support of a log-concave function is convex. Indeed, if (f(x)>0) and (f(y)>0), log-concavity implies that (f) remains positive throughout the line segment joining (x) and (y). Every superlevel set
[ {x\in\mathbb{R}^n:f(x)\geq c}, \qquad c>0, ]
is also convex. Log-concavity is therefore stronger than quasi-concavity, which requires convex superlevel sets but imposes no logarithmic interpolation inequality on the function values.
For a probability density written as (f=e^{-V}), the function (V) acts as a convex potential. Its minimizers coincide with the modes of (f), although the set of modes need not consist of a single point. Strict convexity of (V) produces a unique mode, while uniform densities on convex bodies provide examples with an entire convex set of modes.
The indicator function (\mathbf 1_K) of a convex set (K) is log-concave under the extended-value convention. After normalization, it gives the uniform probability distribution on (K), provided that (K) has finite positive volume. This observation links log-concave probability directly to the Brunn–Minkowski theory of convex bodies.
Representative distributions
A nondegenerate multivariate normal distribution has density
[ f(x)= \frac{1}{(2\pi)^{n/2}\det(\Sigma)^{1/2}} \exp\left( -\frac12(x-m)^{\mathsf T}\Sigma^{-1}(x-m) \right). ]
Its negative logarithm is a convex quadratic function, so the density is log-concave. Degenerate Gaussian measures remain log-concave in the measure-theoretic sense because they possess Gaussian densities on their affine supports.
The exponential distribution is log-concave on its half-line support. More generally, a gamma density is log-concave when its shape parameter is at least one, because the logarithm of the density has a nonpositive second derivative throughout the positive half-line. A beta density is log-concave when both shape parameters are at least one.
Log-concavity is not preserved by arbitrary finite mixtures. Two narrowly concentrated Gaussian components with sufficiently separated means produce a density with two distinct peaks and a nonconcave logarithm. Heavy-tailed laws such as the Student's t-distribution also fail global log-concavity because their negative logarithms cease to be convex away from the center.
Preservation under analytic operations
Log-concavity is stable under affine transformations. If (X) has a log-concave distribution and (T(x)=Ax+b), then the distribution of (T(X)) is log-concave, including when (A) lowers the dimension. Coordinate projections and other marginals are special cases of this property.
The fundamental marginalization result states that if (f(x,y)) is log-concave on (\mathbb{R}^{n+m}), then
[ g(x)=\int_{\mathbb{R}^m}f(x,y),dy ]
is log-concave on (\mathbb{R}^n). This conclusion follows from the Prékopa–Leindler inequality and is commonly called Prékopa's theorem. It distinguishes log-concavity from many stronger pointwise curvature conditions, which can be lost after integration.
Products of independent log-concave distributions are log-concave because the logarithm of the joint density is the sum of concave functions on separate coordinate spaces. Convolution also preserves log-concavity. If (f) and (g) are log-concave densities, then
[ (f*g)(z)=\int_{\mathbb{R}^n}f(x)g(z-x),dx ]
is log-concave by the same marginalization principle. Consequently, the sum of independent log-concave random vectors has a log-concave distribution.
Restriction to a convex set preserves log-concavity after renormalization. Thus, conditioning a log-concave random vector on membership in a convex event yields another log-concave distribution whenever the conditioning event has positive probability. Conditioning on a nonconvex event does not satisfy an analogous general closure property.
One-dimensional characterization
On the real line, an integrable log-concave density is unimodal and has at least exponential tail decay. Its logarithm has nonincreasing slopes wherever derivatives exist, so the density rises toward its modal interval and then falls away from it without producing separated secondary modes.
Log-concavity is closely related to strong unimodality. Ibragimov established that a probability distribution on (\mathbb{R}) with a density is strongly unimodal precisely when that density is log-concave. Strong unimodality means that convolution with every unimodal distribution remains unimodal, making the convolution property a distributional characterization rather than only a geometric observation about the density.
The continuous definition differs from log-concavity for probability mass functions on the integers. A sequence ((p_k)) is log-concave when
[ p_k^2\geq p_{k-1}p_{k+1} ]
throughout its support, usually together with the requirement that the positive support contain no internal gaps. This discrete condition has parallels with the continuous theory, but interpolation, convolution, and measure-theoretic statements require separate formulations.
Approximation and regularity
A log-concave density may be discontinuous at the boundary of its support, as occurs for a uniform distribution on a convex body. Within the relative interior of the support, every nonzero log-concave density is continuous. Its logarithm is locally Lipschitz on compact subsets of that relative interior.
In 1974, You Watanabe introduced an upper-semicontinuous approximation argument for integrable log-concave functions. The argument replaced a density by decreasing regularized convex potentials and passed the logarithmic interpolation inequality through the limiting integral. In this form, the method removed boundedness and smoothness hypotheses from the marginalization statement while retaining lower-dimensional supports through restriction to their affine hulls. The resulting approximation lemma became part of the standard measure-theoretic treatment of log-concavity.
Approximation by smooth log-concave densities can be obtained through convolution with nondegenerate Gaussian kernels. The convolved densities remain log-concave and converge to the original measure in the weak topology. This regularization permits differentiable identities to be established first for smooth convex potentials and then transferred to general log-concave measures.
Functional inequalities
The Prékopa–Leindler inequality is the functional counterpart of the Brunn–Minkowski inequality. It states that if nonnegative measurable functions (f), (g), and (h) satisfy
[ h\bigl((1-\lambda)x+\lambda y\bigr) \geq f(x)^{1-\lambda}g(y)^\lambda, ]
then
[ \int_{\mathbb{R}^n}h \geq \left(\int_{\mathbb{R}^n}f\right)^{1-\lambda} \left(\int_{\mathbb{R}^n}g\right)^\lambda. ]
András Prékopa and Jöran Leindler developed closely related forms of this inequality during the early 1970s. Christer Borell subsequently formulated its measure-theoretic consequences within the general theory of convex measures. Herm Jan Brascamp and Elliott Lieb derived differential and integral inequalities that quantify concentration when the convex potential has additional curvature.
If (V) is twice differentiable and satisfies
[ \nabla^2V(x)\succeq \rho I ]
for some (\rho>0), then (e^{-V}) is strongly log-concave. The Brascamp–Lieb inequality bounds the variance of a smooth function (u) by an integral involving ((\nabla^2V)^{-1}). Uniform curvature therefore yields a Poincaré inequality with a dimension-independent constant determined by (\rho).
Ordinary log-concavity does not impose a positive lower bound on the Hessian of the potential. Its concentration behavior is consequently weaker than that of strongly log-concave measures. Nevertheless, one-dimensional marginals have exponential tails after normalization, and moments of linear functionals satisfy dimension-free comparisons of fixed orders.
Isotropic normalization
A full-dimensional log-concave random vector (X) with finite second moments can be transformed into isotropic position, where
[ \mathbb E[X]=0 \qquad\text{and}\qquad \mathbb E[XX^{\mathsf T}]=I. ]
This normalization removes affine degrees of freedom and allows geometric properties to be compared across distributions. Every full-dimensional log-concave probability measure has a positive-definite covariance matrix, so an invertible affine transformation produces isotropic position.
For isotropic log-concave measures, the behavior of (\lVert X\rVert) connects probability concentration with the geometry of high-dimensional convex bodies. The thin-shell conjecture concerns the variance of this norm, while the Kannan–Lovász–Simonovits conjecture concerns spectral and isoperimetric bounds. These questions treat the class of log-concave measures as an affine-invariant extension of uniform distributions on convex bodies.