space; for the case of convex symmetric functions, see [ 14, Theorem 13.3; .... differential as the concept of derivative, and because the relevant functions.
JOIJRNAL
OF MATHEMATICAL
ANALYSIS
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APPLICATIONS
135, 462475
(1988)
Subdifferentials of Convex Symmetric Functions: An Application of the inequality of Hardy, Littlewood, and P6lya*,+ ANTHONY HORSLEY AND ANDRZEJ J. WROBEL Department
of Economics, London School of Economics and Political Science, Houghton Street, London WC2, United Kingdom Submitted by Ky Fan Received October 30, 1986
I. Schur showed that a differentiable, real-valued, symmetric function defined on a Euclidean space is isotone in the majorisation order if and only if the vector of its partial derivatives at any point is arranged similarly to the vector of the coordinates of that point. This classical result is extended to the case of convex, possibly nonsmooth, symmetric functions defined on rearrangement-invariant function spaces. The result is applied to prove for convex symmetric functibns that rearrangement and subdifferentiation commute. The extreme points of the subdifferentials of the support functions of (i) the set of all functions majorised by a given function, and (ii) the set of all nonnegative functions weakly majorised by a given nonnegative ((3 1988 Academic Press, Inc function, are characterised.
1. INTRODUCTION Real-valued functions which are isotone in the majorisation order of Hardy, Littlewood, and Polya are called Schur functions, after Issai Schur who studied them on Euclidean spaces [21]. For a detailed account of the theory of Schur functions on Euclidean spaces, R”, and some of its applications, see [ 161. Schur functions on R” are symmetric. The converse holds for quasiconvex functions; i.e., every quasi-convex symmetric function on R” is a Schur function [ 16, Chap. 3, Propositions C.2 and C.31. These propositions remain true upon replacing R” by any rearrangementinvariant Banach space of functions defined on a nonatomic measure space; for the case of convex symmetric functions, see [ 14, Theorem 13.3; 3, Theorem 20.31. (The modification required for measures with atoms is * Financial support from BOC Limited during the course of this research is gratefully acknowledged. ’ AMS 1980 Mathematics Subject Classitication (1985): Primary 46605. Secondary 06A10, 46A55, 46E30, 90A14.
462 0022-247X/88 $3.00 CopyrIght @J IY88 by Academic Press, Inc. All rights of reproduction m any form reserved
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given in Section 3 below.) Thus, for quasi-convex functions defined on a rearrangement-invariant function space the terms “symmetric function” and “Schur function” can be used interchangeably (if the underlying measure space is nonatomic). A primary problem in the theory of Schur functions is to characterise them in terms of their derivatives. In the finite-dimensional case this was done by Schur [21] and Ostrowski [17]. Our purpose is to extend this analysis to infinite-dimensional spaces. We use the results in economics where the need to differentiate symmetric functions defined on inlinitedimensional spacesarises in the computation of the marginal costs of commodites for which the production costs depend only on the distribution of output (e.g., over time and/or location and/or states of nature). An instance of this occurs in the economics of electricity, presented in [l&13]. One of our main tasks is to show that the operation of taking, for each point, the gradient (or the subgradients) of a symmetric function at that point and the operation of rearranging points are commuting operations. In infinite-dimensional spaces this presents a problem because measurepreserving mappings are, generally, noninvertible (unlike permutations in the finite-dimensional theory). For functions on finite-dimensional spaces, the symmetry property is a special case of the property of invariance with respect to a group of orthogonal transformations, viz., it is invariance with respect to the permutation group. For the finite-dimensional case, the required commutation result is given, e.g., in [6, Remark 2.23 for smooth functions; and, for nonsmooth, convex functions, a similar result can be deduced from a result on the invariance of the conjugates of groupinvariant functions given in [ 18, Corollary 12.3.1 on p. 1lo] and from the Fenchel inequality [ 18, 512, p. 105]-but note that the group property is essential in this argument. In the infinite-dimensional case, measurepreserving transformations of the underlying measure space form only a semi-group, and, for this reason, the analysis of [ 181 cannot be applied to symmetric functions, although it remains applicable to group-invariant functions. (Other problems of a similar character, some of which appear to be unsolved, are discussed in [22].) However, for convex, symmetric functions it turns out (Theorem 2 below) that the difficulty in proving the commutation result caused by noninvertibility can be circumvented by using an extension of Schur’s differential characterisation to convex functions defined on infinite-dimensional spaces. The extension, given in Theorem 1, is, of course, of interest in itself. The sublinear Schur function which is defined as the composition of the nondecreasing-rearrangement operator and of a linear functional is introduced in Section 2, and it is identified as the support function of an appropriate set, in Formula (1). We term this function a basic convex Schur function because every convex Schur function is the supremum of a family
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of translates of basic convex Schur functions. The subdifferential of a basic convex Schur function is calculated in Theorem 3, and a variant of this result is given in Theorem 4. It is possible first to subdifferentiate a basic convex Schur function at the nondecreasing rearrangement of its argument and then apply the commutation result of Theorem 2. But a difficulty arises in the first step of this procedure because the cone of nondecreasing functions on [0, l] has an empty algebraic interior. A similar obstacle was overcome by J. V. Ryff in his characterisation of the extreme points of the set of all rearrangements of a given function, cf. the remarks in the last paragraph of the introduction to [20]. Therefore, we have chosen to present a proof of Theorem 3 based on Ryffs result, on the converse to the inequality of Hardy, Littlewood, and Polya, and on the formula for the subdifferential of a support function. Finally, note that Theorems 1 and 2 can also be obtained from Theorem 3 by using the representation of convex Schur functions in terms of basic convex Schur functions. Although in this paper the formulation is for Lebesgue spaces, the analysis can be set in the more general rearrangement-invariant spaces introduced by Lorentz and by Luxemburg and Zaanen [ 151. But, since we wish to present the ideas as simply as possible, this is not done here. Also, attention is restricted to convex functions because we wish to use the subdifferential as the concept of derivative, and because the relevant functions in the economic application in [l&13] are convex. However, an analysis is available for wider classesof Schur functions and of symmetric functions. To prevent possible confusion, a few words on the terminology are called for. The term “Schur function” as used in this paper does appear in some recent literature. However, Schur functions used to be, and sometimes still are, called “Schur-convex” even when they are not convex. In [16, p. 541 the term “Schur-increasing” was suggested.In [3, 141 the term “universally Schur-convex” was used to mean a convex Schur function. Convex symmetric functions have also been studied in [7] under the name “Polya functions.”
2. DEFINITIONS, NOTATION, AND PRELIMINARIES The term “function” is used to mean a real-valued mapping. All vector spaces are assumed to be over the real field. Unless the contrary is explicitly stated, the measuresare probability measures.A counting measure is a purely atomic probability measure with atoms of equal weight. Subsets of the interval [0, l] are always taken with the restriction of the Lebesgue measure, mes, normalised so that mes([0, 11) = 1.
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Let (A, &, p) be a measure space. The duality between the Lebesgue spacesL”(p) and Lp’(p), where PE [l, + co] and (p’)-l +p-’ = 1, is written as ( ., . ),; for p = mes the symbol p is suppressed.The pointwise order in L”(p) is written as
