Logarithmically concave function: Difference between revisions

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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]]. 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 (e.g., <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|indicator functions]] of convex sets.