Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures

作者：Milman Emanuel^*; Rotem Liran

来源：Advances in Mathematics, 2014, 262: 867-908.

DOI：10.1016/j.aim.2014.05.023

Elementary proofs of sharp isoperimetric inequalities on a normed space (Er, ll 'ID equipped with a measure p, = w(x)dx so that zoo is homogeneous are provided, along with a characterization of the corresponding equality cases. When p E (0, co] and in addition wP is assumed concave, the result is an immediate corollary of the Borell-Brascamp-Lieb extension of the classical Brunn-Minkowski inequality, providing a new elementary proof of a recent Cabre-Ros-Oton-Serra result. When p E (-1/n, 0), the relevant property turns out to be a novel "q-complemented Brunn-Minkowski" inequality: VA E (0,1) V Borel sets A, B c ir such that /..t(118n \ A), /..4118n \ B) <00, (118n \ (AA + (1 - A)B)) _< (Ap,(118" \ + (1 - A)p,(Rn \ B)")1/, which we show is always satisfied by when wP is homogeneous with 1 = + n- in particular, this is satisfied by q 7,

• 出版日期2014-9-10