The Orlicz-Brunn-Minkowski theory: A general framework, additions, and inequalities

Abstract

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 L p -Brunn-Minkowski theory, analogous to the way that Orlicz spaces generalize L p spaces. For appropriate convex functions ϕ : [0, ∞) m → [0, ∞), a new way of combining arbitrary sets in R n is introduced. This operation, called Orlicz addition and denoted by + ϕ , has several desirable properties, but is not associative unless it reduces to L p 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 first 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 L p -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 L p -Brunn-Minkowski inequality. New Orlicz-Minkowski inequalities are obtained that generalize the L p -Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequality of Lutwak, Yang, and Zhang, and in fact these two inequalities together are shown to split the classical Brunn-Minkowski inequality.

Symmetrization in geometry

“…We have (11) that (10) holds. Conversely, assume that (10) holds and let x ∈ L be of maximal distance from o.…”

“…In particular, [9,Corollary 8.4] (see also [1]) classifies central symmetrization, defined by (3) below. Another new symmetrization, M-symmetrization, is introduced in Section 4 and employs the notion of M-addition studied in [9,10]. There are many other symmetrization processes in geometry, such as those leading to the fundamental notions of projection body, intersection body, and centroid body (see [8,28]).…”

“…Section 4, on projection covariant symmetrizations, may be regarded as a sequel to the investigation of Gardner, Hug, and Weil [9,10] into additions, such as Minkowski and L p addition, in convex geometry. In fact, certain symmetrizations, such as central symmetrization, result from adding a set to its reflection in the origin.…”

Operations between functions

“…, h Km ) allows an operation between compact convex sets to be transferred to one between their support functions, and vice versa. In this context, [13,Lemma 3.2] says that pointwise operations between support functions of o-symmetric compact convex sets, with associated positively homogeneous function F , correspond precisely to projection covariant operations between the sets themselves.…”

“…The first is the introduction in Section 8 of Orlicz addition between functions. This is motivated by recent developments in the Brunn-Minkowski theory, the heart of convex geometry, in which Orlicz addition of sets, a generalization of L p addition of sets, was recently discovered; see [13,14,30,31]. Orlicz addition, denoted by + ϕ and defined by (8.2) below, is an operation + ϕ : Φ(A) m → Φ(A) in several useful instances, for example when Φ(A) is the class of nonnegative Borel or nonnegative continuous functions on A, or Cvx…”