Logarithmically concave function

A function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{+}}$ is logarithmically concave (or log-concave for short), if its natural logarithm ${\displaystyle \ln(f(x))}$, is concave. Note that we allow here concave functions to take value ${\displaystyle -\infty }$. Every concave function is log-concave, however the reverse does not necessarily hold: an example is the function ${\displaystyle e^{-x^{2}}}$ which is log-concave (${\displaystyle -x^{2}}$ is a concave function of ${\displaystyle x}$) but is not concave for ${\displaystyle |x|>1/{\sqrt {2}}}$.

Examples of log-concave functions are the indicator functions of convex sets and the Gaussian function.

In parallel, a function is log-convex if its natural log is convex.

See also

• Convex function
• Logarithmically concave measure