Logarithmically concave function

In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality f ( θ x + ( 1 − θ ) y ) ≥ f ( x ) θ f ( y ) 1 − θ {\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }} for all x,y ∈ dom f and 0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is, log ⁡ f ( θ x + ( 1 − θ ) y ) ≥ θ log ⁡ f ( x ) + ( 1 − θ ) log ⁡ f ( y ) {\displaystyle \log f(\theta x+(1-\theta )y)\geq \theta \log f(x)+(1-\theta )\log f(y)} for all x,y ∈ dom f and 0 < θ < 1.

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