Logarithmically concave function: Difference between revisions

Revision as of 15:35, 28 July 2010

Revision as of 19:23, 21 February 2011

Line 1:

{{Unreferenced|date=December 2009}}

In [[mathematics]], a function <math>{{nowrap|''f'' : \'''R^'''^''n'' \to→ \'''R^'''<sup>+</mathsup>}} is '''logarithmically concave''' (or '''log-concave''' for short), if its [[natural logarithm]] <math>\{{nowrap|ln(&thinsp;''f''(''x''))</math>}}, is [[concave function|concave]]. This means that it must be:

: <math>

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

</math>

Note that we allow here concave functions to take value -&minus;∞.

:<math>

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

\geq

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

\qquad

\theta \in [0, 1]

</math>

Note that we allow here concave functions to take value -∞.

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