 Log-concave probability density

A probability density function $f$ is log-concave if, for every $0<t<1$,Equivalently, the extended-valued function $\log f$, with $\log 0=-\infty$, is a concave function. In particular, the positive-density region must be a convex set. Stating concavity only on an arbitrarily disconnected support misses this requirement.

For example, the Gaussian distribution has log density equal to a constant minus a positive quadratic, hence is log-concave. A uniform distribution on a convex set of finite positive volume is also log-concave. But density $1/2$ on $[-2,-1]\cup [1,2]$ is not: its values at $-3/2$ and $3/2$ are positive while its value at their midpoint is zero.

At an interior point $x_0$ where $\log f$ is differentiable, a supporting tangent gives $\log f(x)\le \log f(x_0)+\nabla \log f(x_0)\cdot (x-x_0)$. Exponentiating produces an upper envelope used in rejection sampling. The envelope must also have finite integral on the sampling region to serve as a normalized proposal probability density function.