= Log-concave probability density
A <probability density function> $f$ is log-concave if, for every $0<t<1$,
$$
f(tx+(1-t)y)\geq f(x)^t f(y)^{1-t}.
$$
Equivalently, the extended-valued <function> $\log f$, with $\log0=-\infty$, 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 <probability density function>[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 <probability density function>[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 $\log f(x)\leq\log f(x_0)+\nabla\log f(x_0)\cdot(x-x_0)$. 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>.
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