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$L_p$-Brunn-Minkowski inequality for $p\in (1-\frac{c}{n^{\frac{3}{2}}}, 1)$

Journal Article


Abstract


  • Kolesnikov-Milman [9] established a local $L_p$-Brunn-Minkowski inequality

    for $p\in(1-c/n^{\frac{3}{2}},1).$ Based on their local uniqueness results for

    the $L_p$-Minkowski problem, we prove in this paper the (global)

    $L_p$-Brunn-Minkowski inequality. Two uniqueness results are also obtained: the

    first one is for the $L_p$-Minkowski problem when $p\in (1-c/n^{\frac{3}{2}},

    1)$ for general measure with even positive $C^{\alpha}$ density, and the second

    one is for the Logarithmic Minkowski problem when the density of measure is a

    small $C^{\alpha}$ even perturbation of the uniform density.

Publication Date


  • 2018

Web Of Science Accession Number


Abstract


  • Kolesnikov-Milman [9] established a local $L_p$-Brunn-Minkowski inequality

    for $p\in(1-c/n^{\frac{3}{2}},1).$ Based on their local uniqueness results for

    the $L_p$-Minkowski problem, we prove in this paper the (global)

    $L_p$-Brunn-Minkowski inequality. Two uniqueness results are also obtained: the

    first one is for the $L_p$-Minkowski problem when $p\in (1-c/n^{\frac{3}{2}},

    1)$ for general measure with even positive $C^{\alpha}$ density, and the second

    one is for the Logarithmic Minkowski problem when the density of measure is a

    small $C^{\alpha}$ even perturbation of the uniform density.

Publication Date


  • 2018

Web Of Science Accession Number