Logarithmically concave function

In <a href="/facts/Convex_analysis/SGhUNLMK">convex analysis</a>, a <a href="/facts/Non-negative/z0xollg1">non-negative</a> function f : Rn → R+ is logarithmically concave (or log-concave for short) if its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> is a <a href="/facts/Convex_set/vdAuJRJl">convex set</a>, and if it satisfies the inequality

f
        (
        θ
        x
        +
        (
        1
        −
        θ
        )
        y
        )
        ≥
        f
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          )
          
            θ
          
        
        f
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        y
        
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            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 <a href="/facts/Logarithm/QeXaV97q">logarithm</a> of the function, log ∘ f, is <a href="/facts/Concave_function/HxVEBkce">concave</a>; that is,

log
        ⁡
        f
        (
        θ
        x
        +
        (
        1
        −
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        )
        y
        )
        ≥
        θ
        log
        ⁡
        f
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        +
        (
        1
        −
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        log
        ⁡
        f
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    {\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.
Examples of log-concave functions are the 0-1 <a href="/facts/Indicator_function/QEuh04NM">indicator functions</a> of convex sets (which requires the more flexible definition), and the <a href="/facts/Gaussian_function/5bG8Kpp9">Gaussian function</a>.
Similarly, a function is <a href="/facts/Log-convex/1fyzx7Ig">log-convex</a> if it satisfies the reverse inequality

f
        (
        θ
        x
        +
        (
        1
        −
        θ
        )
        y
        )
        ≤
        f
        (
        x
        
          )
          
            θ
          
        
        f
        (
        y
        
          )
          
            1
            −
            θ
          
        
      
    
    {\displaystyle f(\theta x+(1-\theta )y)\leq f(x)^{\theta }f(y)^{1-\theta }}

for all x,y ∈ dom f and 0 < θ < 1.

Logarithmically concave function open-in-new

Logarithmically concave function