Convex measure: Difference between revisions

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Revision as of 00:05, 24 May 2018

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:<math>M_{s, \lambda}(u, v) = \begin{cases} (\lambda u^s + (1 - \lambda) v^{s})^{1/s} & \text{if } - \infty < s < 0, \\ \min(u, v) & \text{if } s = - \infty, \\ u^{\lambda} v^{1- \lambda} & \text{if } s = 0. \end{cases}</math>

For subsets ''A'' and ''B'' of ''X'', we write

:<math>\lambda A + (1 - \lambda) B = \{ \lambda x + ( 1 - \lambda ) By \mid x \in A, y \in B \}</math>

for their [[Minkowski sum]]. With this notation, the measure ''&mu;'' is said to be '''''s''-convex'''<ref name="Borell1974"/> if, for all Borel-measurable subsets ''A'' and ''B'' of ''X'' and all 0 &le; ''&lambda;'' &le; 1,

:<math>\mu(\lambda A + (1 - \lambda) B) \geq M_{s, \lambda}(\mu(A), \mu(B)).</math>