The dual Orlicz–Brunn–Minkowski theory ☆

By Richard J. Gardner, Daniel Hug, Wolfgang H. Weil, Deping Ye. 2014

141 citations recorded by the source corpus.

This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz- Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is derived and a corresponding dual Orlicz-Minkowski inequality proved. The inequalities proved yield as special cases the precise duals of the conjectured log-Brunn-Minkowski and log-Minkowski inequalities of Boroczky, Lutwak, Yang, and Zhang. A new addition of star sets called radial M-addition is also introduced and shown to relate to the radial Orlicz addition.

Subfields: Applied Mathematics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/70af7785344bc8733da78e6cb5af4670bfbc8b1f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Yan-nan Liu
2. Sudan Xing
3. Baocheng Zhu
4. Jian Lu
5. Deping Ye
6. Yiming Zhao

In Pharmacology

1. Sudan Xing
2. Baocheng Zhu
3. Deping Ye

Citations in the corpus

• A flow method for the dual Orlicz–Minkowski problem (2020, background): “ometry, the Orlicz-Brunn-Minkowski theory and the dual Orlicz-Brunn-Minkowski theory are developing rapidly in recent years. They have attracted great attention from many researchers, see for example [11, 20, 22, 27, 29, 30, 32, 33, 34, 36, 43, 45, 51, 53, 55, 56, 58, 60, 63, 66, 69] and references ”
• The Dual Orlicz–Minkowski Problem (2017, methodology): “(t) = ∞ and limt→∞ ϕi(t) = 0, i= 1,2, the function ρK+e ϕ,εL can also be defined by formula (3.3). If ϕ′ 1(1), the derivative of ϕ1 at 1, exists and is nonzero, the following variational formula holds [13, 58] : ϕ′ 1(1) lim ε→0+ V(K+eϕ,εL)−V(K) ε = Z Sn−1 ϕ2 ρL(u) ρK(u) ρK(u)ndu. 6 Motivated by thi”
• The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems (2017, unknown): “It follows from Jensen’s inequality (see [17]), Cp(Ω) > 0 and the convexity of φ that Cp,φ(Ω,Ω1) = p− 1 n− p ∫”
• Dual Orlicz–Brunn–Minkowski theory: Dual Orlicz Lϕ affine and geominimal surface areas (2014, background): “holds if and only if Kis an origin-symmetric Euclidean ball. 3.2 Orlicz Lφ-dual mixed volume Let φ : (0,∞) → (0,∞) be a continuous function. The Orlicz Lφ-dual mixed volume for K,L∈ S 0 was defined in [10, 55] by Ve φ(K,L) = 1 n Z Sn−1 φ ρL(u) ρK(u) [ρK(u)]ndσ(u). Clearly, if L= λKfor some λ>0, ”