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The following are among the properties of log-concave distributions: If a density is log-concave, so is its cumulative distribution function (CDF). If a multivariate density is log-concave, so is the marginal density over any subset of variables. The sum of two independent log-concave random variables is log-concave. This follows from the fact ...
The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave. The restriction of the Lebesgue measure to any convex set is also log-concave.. By a theorem of Borell, [2] a probability measure on R^d is log-concave if and only if it has a density with respect to the Lebesgue measure on some affine hyperplane, and this density is a logarithmically concave function.
The rows of Pascal's triangle are examples for logarithmically concave sequences. In mathematics, a sequence a = (a 0, a 1, ..., a n) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if a i 2 ≥ a i−1 a i+1 holds for 0 < i < n.
The logarithm function () = is concave on its domain (,), as its derivative is a strictly decreasing function. Any affine function f ( x ) = a x + b {\displaystyle f(x)=ax+b} is both concave and convex, but neither strictly-concave nor strictly-convex.
Log-concave may refer to: Logarithmically concave function; Logarithmically concave measure; Logarithmically concave sequence This page was last edited on 22 ...
A logarithmically convex function f is a convex function since it is the composite of the increasing convex function and the function , which is by definition convex.However, being logarithmically convex is a strictly stronger property than being convex.
A log–log plot of y = x (blue), y = x 2 (green), and y = x 3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1. Comparison of linear, concave, and convex functions when plotted using a linear scale (left) or a log scale (right).
Because the logarithm function is concave, (+ ()) + = + = with the equality holding if and only if =. Young's inequality follows by exponentiating. Yet another proof is to first prove it with b = 1 {\displaystyle b=1} an then apply the resulting inequality to a b q {\displaystyle {\tfrac {a}{b^{q}}}} .