Logarithmically concave measure

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Borel_measure/8cYR8Ysw">Borel measure</a> μ on n-<a href="/facts/Dimension/0ktmUyac">dimensional</a> <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 is called logarithmically concave (or log-concave for short) if, for any <a href="/facts/Compact_set/d0cgXJH7">compact subsets</a> A and B of 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 and 0 < λ < 1, one has

μ
        (
        λ
        A
        +
        (
        1
        −
        λ
        )
        B
        )
        ≥
        μ
        (
        A
        
          )
          
            λ
          
        
        μ
        (
        B
        
          )
          
            1
            −
            λ
          
        
        ,
      
    
    {\displaystyle \mu (\lambda A+(1-\lambda )B)\geq \mu (A)^{\lambda }\mu (B)^{1-\lambda },}

where λ A + (1 − λ) B denotes the <a href="/facts/Minkowski_sum/dEQfoJ1A">Minkowski sum</a> of λ A and (1 − λ) B.

Logarithmically concave measure open-in-new

Logarithmically concave measure