logconcave

Logconcave, or log-concave, describes a property of nonnegative functions that is central in convex analysis and its applications. A function f defined on a convex domain D ⊆ R^n is log-concave if for all x, y in D and all t in [0, 1], f(tx + (1−t)y) ≥ f(x)^t f(y)^(1−t). If f is strictly positive on D, this is equivalent to saying that log f is a concave function on D.

The log-concavity condition has several useful consequences. The set where f is positive is convex, and the level sets {x ∈ D : f(x) ≥ α} are convex for all α > 0. Equivalently, log f being concave implies that f is unimodal in a precise sense and tends to concentrate mass around its mode. The concept extends naturally to probability densities, where log-concave densities have desirable stability and concentration properties.

Examples of log-concave functions include the Gaussian density, the Laplace (double-exponential) density, the exponential distribution on the positive real line, and the uniform distribution on any convex set. The product of log-concave functions is log-concave, and the convolution of log-concave functions is also log-concave. Marginals of log-concave joint densities remain log-concave, a consequence of the Prékopa–Leindler inequality. By contrast, mixtures of well-separated modes are generally not log-concave.

Log-concavity appears in optimization, statistics, economics, and machine learning. It supports efficient optimization of log-concave objective functions, guarantees unimodality and strong concentration in probabilistic models, and underpins many results on sampling and inference for log-concave distributions.