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. This means that it must be:

${\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }\;\;\;\;\;\theta \in [0,1]}$

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.

A log-concave function is also quasi-concave.

Operations preserving the log-concavity

• product (the product of log-concave functions is a log-concave function. Notice this is *not* true for the sum of log-convex functions)
• integration:

${\displaystyle f(x,y):{\mathcal {R}}^{n+m}\rightarrow {\mathcal {R}}\;\;{\text{is log concave}}\;\;\Rightarrow \int f(x,y)dy\;\;{\text{is log concave}}}$ 

(this imply that convolution is an operation preserving log-concavity).

See also

• log-convex function
• Convex function
• Logarithmically concave measure