Convex measure: Difference between revisions

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Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class.

The convexity of a measure ''&mu;'' on ''n''-dimensional [[Euclidean space]] '''R'''^''n'' in the sense above is closely related to the convexity of its [[probability density function]].<ref name="Borell1975"/> Indeed, ''&mu;'' is ''s''-convex if and only if there is an [[absolutely continuous measure]] ''&nu;'' with probability density function ''&rho;'' on some '''R'''^''k'' so that ''&mu;'' is the [[Pushforward measure|push-forward]]{{dn|date=August 2017}} on ''&nu;'' under a [[linear map|linear or affine map]] and <math>e_{s, k} \circ \rho \colon \mathbb{R}^{k} \to \mathbb{R}</math> is a [[convex function]], where

:<math>e_{s, k}(t) = \begin{cases} t^{s / (1 - s k)} & \text{if } -\infty < s < 0 \\ t^{-1/k} & \text{if } s = - \infty, \\ - \log t & \text{if } s = 0.\end{cases}</math>