A unified treatment for $L_p$ Brunn-Minkowski type inequalities

Abstract

A unified approach used to generalize classical Brunn-Minkowski type inequalities to L p Brunn-Minkowski type inequalities, called the L p transference principle, is refined in this paper. As illustrations of the effectiveness and practicability of this method, several new L p Brunn-Minkowski type inequalities concerning the mixed volume, moment of inertia, quermassintegral, projection body and capacity are established.2010 Mathematics Subject Classification: 52A40.

Inequalities on the (p,q)-mixed volume involving Lp centroid bodies and Lp intersection bodies

“…and, according to the condition that the equality holds in ( 14), we see that the equality holds in (22) if and only if ρ L 1 is proportional to ρ L 2 , or equivalently, L 1 and L 2 are dilations.…”

“…The classical Brunn-Minkowski theory is mainly concerned with the analogues and generalizations of the Brunn-Minkowski inequality for geometric quantities. In 2018, Zou and Xiong [22] has formulated the L p transference principle, and the L p Brunn-Minkowski type inequalities they established, characterize the concavity of existing functionals, in terms of the L p addition of convex bodies. This paper first focus on establishing Brunn-Minkowski type inequalities, in terms of the dual Blaschke addition (also called radial Blaschke sum) introduced by Guo-Jia [7].…”

The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems

“…As ϕ(x) = m i=1 x q j for q > 1 is strictly convex, equality holds if and only if Ω i is dilate of Ω j for all 1 ≤ i < j ≤ m. This has been proved by Zou and Xiong in [54] with a different approach. Now let us consider the linear Orlicz addition of Ω 1 , · · · , Ω m ∈ C 0 .…”

Lp Minkowski problem for electrostatic $\mathfrak{p}$-capacity

“…Then we will use it to extend the p-capacitary Brunn-Minkowski inequality to the L p stage. It is interesting that the L p Brunn-Minkowski type inequality for p-capacity was previously established in [77] by the authors' L p transference principle.…”