Brunn–Minkowski inequality for mixed intersection bodies

Abstract

Hölder's inequality states that x p y q − −x, y ≥ 0 for any (x, y) ∈ L p (Ω) × L q (Ω) with 1/p + 1/q = 1. In the same situation we prove the following stronger chains of inequalities, where z = y|y| q−2 : x p y q − −x, y ≥ (A similar result holds for complex valued functions with Re(x, y) substituting for x, y. We obtain these inequalities from some stronger (though slightly more involved) ones.

On Blaschke-Minkowski homomorphisms

“…These inequalities for mixed projection bodies were established by Lutwak [7]. Moreover, Brunn-Minkowski type inequality for mixed intersection bodies were established in [13,14]. Theorem 2.2 Let : K n → K n be a Blaschke-Minkowski homomorphism.…”

On Reverse Minkowski-Type Inequalities

“…Here,Ṽ 1 (K, L) is the dual mixed volume of the star bodies K and L. A more general version of the dual Minkowski inequality states (see [34]): If K and L are star bodies in R n , and 0 ≤ i < n − 1, thenW…”

Dual Orlicz mixed geominimal surface area

“…Subsequently, through the efforts of many scholars, the L p -BrunnMinkowski theory (see [11]) has been put forward and some classical inequalities were proved, such as affine isoperimetric inequality (see [13]). Here, we provide some reference data for relevant applications (see, e.g., [23,28,32,34,[36][37][38]). …”