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

Abstract

Elementary proofs of sharp isoperimetric inequalities on a normed space (Rn,‖⋅‖) equipped with a measure μ=w(x)dx so that wp is homogeneous are provided, along with a characterization of the corresponding equality cases. When p∈(0,∞] 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 Cabré–Ros-Oton–Serra result. When p∈(−1/n,0), the relevant property turns out to be a novel “q-complemented Brunn–Minkowski” inequality:∀λ∈(0,1)∀ Borel sets A,B⊂Rnsuch thatμ(Rn∖A),μ(Rn∖B)<∞,μ⁎(Rn∖(λA+(1−λ)B))≤(λμ(Rn∖A)q+(1−λ)μ(Rn∖B)q)1/q, which we show is always satisfied by μ when wp is homogeneous with 1q=1p+n; in particular, this is satisfied by the Lebesgue measure with q=1/n. This gives rise to a new class of measures, which are “complemented” analogues of the class of convex measures introduced by Borell, but which have vastly different properties. The resulting isoperimetric inequality and characterization of isoperimetric minimizers extends beyond the recent results of Cañete–Rosales and Howe. The isoperimetric and Brunn–Minkowski type inequalities also extend to the non-homogeneous setting, under a certain log-convexity assumption on the density. Finally, we obtain functional, Sobolev and Nash-type versions of the studied inequalities.