Logarithmically concave function: Difference between revisions

Revision as of 00:20, 28 June 2010

Revision as of 15:35, 28 July 2010

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{{Unreferenced|date=December 2009}}

AIn [[mathematics]], 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]]. This means that it must be:

:<math>

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

A log-concave function is also [[Quasi-concave_function | quasi-concave]].

==Properties==

* A log-concave function is also [[Quasi-concave_function | quasi-concave]].

* Every concave function is log-concave, however the reverse does not necessarily hold. An example is the function

* Every nonnegative concave function is log-concave, however the reverse does not necessarily hold. An example is the function

:<math>f(x) = e^{-x^2 / 2}</math>

which is log-concave since