Logarithmically concave function: Difference between revisions

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{{Unreferenced|date=December 2009}}

A function <math>f : \R^n \to \R^+</math> is '''logarithmically concave''' (or '''log-concave''' for short), if its [[natural logarithm]] <math>\ln(f(x))</math>, is [[concave function|concave]]. NoteThis means that weit allowmust here concave functions to take value <math>-\infty</math>. Every concave function is log-concave, however the reverse does not necessarily holdbe: an example is the function <math>e^{-x^2}</math> which is log-concave (<math>-x^2</math> is a concave function of <math>x</math>) but is not concave for <math>|x| > 1/\sqrt{2}</math>.

<math>

f( \theta x + (1 - \theta) y )

\geq

f( x )^{\theta} f( y )^{1 - \theta}

\;\;\;\;\;

\theta \in [0, 1]

</math>

Note that we allow here concave functions to take value <math>-\infty</math>. Every concave function is log-concave, however the reverse does not necessarily hold: an example is the function <math>e^{-x^2}</math> which is log-concave (<math>-x^2</math> is a concave function of <math>x</math>) but is not concave for <math>|x| > 1/\sqrt{2}</math>.

Examples of log-concave functions are the [[indicator function]]s 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_function | 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:

<math>

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

\Rightarrow

\int f(x,y) dy \;\; \text{is log concave}

</math>

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

==See also==

*[[log-convex | log-convex function]]

*[[Convex function]]

*[[Logarithmically concave measure]]