THE ORLICZ-BRUNN-MINKOWSKI THEORY: A GENERAL FRAMEWORK, ADDITIONS, AND INEQUALITIES

By Richard J. Gardner, Daniel Hug, Wolfgang H. Weil. 2013

264 citations recorded by the source corpus.

The Orlicz-Brunn-Minkowski theory, introduced by Lutwak, Yang, and Zhang, is a new extension of the classical Brunn-Minkowski theory. It represents a generalization of the Lp-Brunn-Minkowski theory, analogous to the way that Orlicz spaces generalize Lp spaces. For appropriate convex functions ' : (0;1) m ! (0;1), a new way of combining arbitrary sets in R n is introduced. This operation, called Orlicz addition and denoted by +', has several desirable properties, but is not associative unless it reduces to Lp addition. A general framework is introduced for the Orlicz-Brunn-Minkowski theory that includes both the new addition and previously introduced concepts, and makes clear for the rst time the relation to Orlicz spaces and norms. It is also shown that Orlicz addition is intimately related to a natural and fundamental generalization of Minkowski addition called M-addition. The results obtained show, roughly speaking, that the Orlicz-Brunn-Minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the Lp-Brunn-Minkowski theory. Inequalities of the Brunn-Minkowski type are obtained, both for M-addition and Orlicz addition. The new Orlicz-Brunn-Minkowski inequality implies the Lp-Brunn-Minkowski in- equality. New Orlicz-Minkowski inequalities are obtained that generalize the Lp-Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequal- ity of Lutwak, Yang, and Zhang, and in fact these two inequalities together are shown to split the classical Brunn-Minkowski inequality.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/9d33082ea7fdae0fb6d190785ea19c14969bc59b

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In Applied Mathematics

1. Yiming Zhao
2. Deping Ye
3. Jian Lu
4. Yong Huang
5. Sudan Xing
6. Yan-nan Liu
7. Baocheng Zhu
8. Dongmeng Xi

Citations in the corpus

• Dual Orlicz-Brunn-Minkowski theory (2014, unknown): “This theory is far more general than the Lp Brunn–Minkowski theory, we refer the reader to [6,16,24,26,29,32,40,43,44,46,53, 57].”
• Remarks on the conjectured log-Brunn–Minkowski inequality (2013, background): “stated explicitly) that inequalities (1), (2) are actually equivalent in the class of symmetric convex bodies in Rn, for 0 < λ,p < 1 and that (3) would imply (1), (2) for all p > 0 (see also [18] for another application of the conjectured log-Brunn-Minkowksi inequality). Moreover, the autho”
• Orlicz–John ellipsoids (2014, methodology): “Very recently, one essential obstacle in the development of Orlicz Brunn–Minkowski theory, what is the lack of a notion corresponding to Lp addition, has been smoothed by Gardner, Hug and Weil [14,15].”
• The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems (2017, background): “εϕ2 hL(u) hK+ϕ,εL(u) , (2.6) where ε>0, K,L∈ K0, and ϕ1,ϕ2 ∈ Φ1. If (ϕ1)′ l(1), the left derivative of ϕ1 at t= 1, exists and is positive, then the Lϕ2 mixed volume of K,L∈ K0 can be defined by [16, 43, 48] Vϕ2(K,L) = (ϕ1)′ l(1) n · d dε |K+ϕ,ε L| ε=0+ = 1 n Z Sn−1 ϕ2 hL(u) hK(u) hK(u)dS(K,u).”
• The dual Orlicz–Brunn–Minkowski theory ☆ (2014, methodology): “ch x ∈ (0,∞)m. We also denote by Φ (1) m and Ψ (1) 1 the classes of functions in Φm and Ψ 1, respectively, such that ϕ(ej) = 1 for j = 1,...,m. We caution the reader that similar notation was used in [16] for different classes of functions. The prototype function is ϕ(x 1,...,xm) = x p 1+···+xp m, wh”
• Dual Orlicz–Brunn–Minkowski theory: Dual Orlicz Lϕ affine and geominimal surface areas (2014, unknown): “for estimates on c). 3 Orlicz φ-mixed volume and its dual 3.1 Orlicz φ-mixed volume Let φ: (0,∞) → (0,∞) be a continuous function. Define the Orlicz φ-mixed volume Vφ(K,Q) of convex bodies K,Q∈ K 0 by [9, 47, 51] Vφ(K,Q) = 1 n Z Sn−1 φ hQ(u) hK(u) hK(u)dS(K,u), 3 where S(K,·) on Sn−1 is the surface”
• 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): “ the Orlicz-Brunn-Minkowski theory of convex bodies. This new theory started from the works of Lutwak, Yang and Zhang [35, 36] and Ludwig [27], and was greatly pushed forward by Gardner, Hug and Weil [12] and independently by Xi, Jin and Leng [46], due to the discovery of the Orlicz addition of conv”
• The Minkowski problem in Gaussian probability space (2020, result): “See also [22,36,40,48,71,78] for additional results and contributions in the Orlicz extension of the classical Brunn-Minkowski theory.”