| Popis: |
In this paper we address the following question: given a measure μ on R n , does there exist a constant C > 0 such that, for any m-dimensional subspace H ⊂ R n and any convex body K ⊂ R n , the following sectional Rogers-Shephard type inequality holds: μ ( ( K − K ) ∩ H ) ≤ C sup y ∈ R n μ ( ( y − K ) ∩ H ) ? We show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant C ( n , m ) = ( n + m m ) . We also prove marginal inequalities of the Rogers-Shephard type for ( 1 / s ) -concave when 0 ≤ s ∞ , and logarithmically concave functions. |
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