Outer Γ-Convexity in Vector Spaces

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ? X if for all x ₀, x ₁ ? S there exists a closed subset Λ ? [0,1] such that {x _λ | λ ? Λ} ? S and [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ, where x _λ: = (1 ? λ)x ₀ + λ x ₁. A real-valued function f:D → ? defined on some convex D ? X is called outer Γ-convex if for all x ₀, x ₁ ? D there exists a closed subset Λ ? [0,1] such that [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ and f(x _λ) ≤ (1 ? λ)f(x ₀) + λ f(x ₁) holds for all λ ? Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.     

E-Convex Sets, E-Convex Functions, and E-Convex Programming

A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.     

Reciprocally convex functions

We say that f is reciprocally convex if x?f(x) is concave and x?f(1/x) is convex on (0,+∞). Reciprocally convex functions generate a sequence of quasi-arithmetic means, with the first one between harmonic and arithmetic mean and others above the arithmetic mean. We present several examples related to the gamma function and we show that if f is a Stieltjes transform, then −f is reciprocally convex. An application in probability is also presented.     

General decomposition theorems form-convex sets in the plane

A setS inR ^dis said to bem-convex,m≧2, if and only if for everym distinct points inS, at least one of the line segments determined by these points lies inS. Clearly any union ofm?1 convex sets ism-convex, yet the converse is false and has inspired some interesting mathematical questions: Under what conditions will anm-convex set be decomposable intom?1 convex sets? And for everym≧2, does there exist aσ(m) such that everym-convex set is a union ofσ(m) convex sets? Pathological examples convince the reader to restrict his attention to closed sets of dimension≦3, and this paper provides answers to the questions above for closed subsets of the plane. IfS is a closedm-convex set in the plane,m ≧ 2, the first question may be answered in one way by the following result: If there is some lineH supportingS at a pointp in the kernel ofS, thenS is a union ofm ? 1 convex sets. Using this result, it is possible to prove several decomposition theorems forS under varying conditions. Finally, an answer to the second question is given: Ifm≧3, thenS is a union of (m?1)³2 ^m?3 or fewer convex sets.     

On the convexity of the multiplicative potential and penalty functions and related topics

It is well known that a function f of the real variable x is convex if and only if (x,y)→yf(y ^-1 x),y>0 is convex. This is used to derive a recursive proof of the convexity of the multiplicative potential function. In this paper, we obtain a conjugacy formula which gives rise, as a corollary, to a new rule for generating new convex functions from old ones. In particular, it allows to extend the aforementioned property to functions of the form (x,y)→g(y)f(g(y)^-1 x) and provides a new tool for the study of the multiplicative potential and penalty functions. Received: June 3, 1999 / Accepted: September 29, 2000?Published online January 17, 2001     

Transitivity of the norm on Banach spaces¶ having a Jordan structure

We study transitivity conditions on the norm of JB ^*-triples, C ^*-algebras, JB-algebras, and their preduals. We show that, for the predual X of a JBW ^*-triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB ^*-triple with transitive norm is a Hilbert space. We extend to arbitrary C ^*-algebras previously known characterizations of transitivity [20] and convex transitivity [36] of the norm on commutative C ^*-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB-algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB-algebra with almost transitive norm is isomorphic to the field of real numbers. Received: 9 June 1999 / Revised version: 20 February 2000     

Nonemptiness of Approximate Subdifferentials of Midpoint δ-Convex Functions

ABSTRACT

For some given positive δ, a function f:D ? X → ? is called midpoint δ-convex if it satisfies the Jensen inequality f[(x ₀ + x ₁)/2] ≤ [f(x ₀) + f(x ₁)]/2 for all x ₀, x ₁ ∈ D satisfying ‖x ₁ ? x ₀‖ ≥ δ (Hu, Klee, and Larman, SIAM J. Control Optimiz. Vol. 27, 1989). In this paper, we show that, under some assumptions, the approximate subdifferentials of midpoint δ-convex functions are nonempty.     

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak^*-continuous on dual spaces. In particular, if X is a subspace of a C^*-algebra A, and if a∈A satisfies aX⊂X, then we show that the function x?ax on X is automatically weak^* continuous if either (a) X is a dual operator space, or (b) a^*X⊂X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C^*-subalgebra. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W^*-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.     

On directional convexity

Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R ^d → R is called D-convex, where D is a set of vectors in R ^d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R ^d, denoted by co^D(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R ² where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co ^D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R ³ (where D is the orthonormal basis of R ³), we construct arbitrarily large finite sets A with co ^D(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices. This research was supported by Charles University Grants No. 158/99 and 159/99.     

On the Minimizing Trajectory of Convex Functions with Unbounded Level Sets

We consider a convex function f(x) with unbounded level sets. Many algorithms, if applied to this class of functions, do not guarantee convergence to the global infimum. Our approach to this problem leads to a derivation of the equation of a parametrized curve x(t), such that an infimum of f(x) along this curve is equal to the global infimum of the function on ⁿ .We also investigate properties of the vectors of recession, showing in particular how to determine a cone of recession of the convex function. This allows us to determine a vector of recession required to construct the minimizing trajectory.     

Biconvex sets and optimization with biconvex functions: a survey and extensions

The problem of optimizing a biconvex function over a given (bi)convex or compact set frequently occurs in theory as well as in industrial applications, for example, in the field of multifacility location or medical image registration. Thereby, a function is called biconvex, if f(x,y) is convex in y for fixed x∈X, and f(x,y) is convex in x for fixed y∈Y. This paper presents a survey of existing results concerning the theory of biconvex sets and biconvex functions and gives some extensions. In particular, we focus on biconvex minimization problems and survey methods and algorithms for the constrained as well as for the unconstrained case. Furthermore, we state new theoretical results for the maximum of a biconvex function over biconvex sets. J. Gorski and K. Klamroth were partially supported by a grant of the German Research Foundation (DFG).     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

Weighted inequalities for iterated maximal functions in Orlicz spaces

Let M be the classical Hardy‐Littlewood maximal operator. The object of our investigation in this paper is the iterated maximal function M^kf(x) = M(M^k?1f) (x) (k ≥ 2). Let Φ be a φ‐function which is not necessarily convex and Ψ be a Young function. Suppose that w is an A′_∞ weight and that k is a positive integer. If there exist positive constants C₁ and C₂ such that ((I)) then there exist positive constants C₃ and C₄ such that ((II)) where the functions a(t) and b(t) are the right derivatives of Φ(t) and Ψ(t), respectively. Conversely, if w is an A₁ weight, then (II) implies (I). Another necessary and sufficient condition will be given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On holomorphic domination, I

Let X be a separable Banach space and u:X→R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h:X→Z with u(x)<‖h(x)‖ for x∈X. As a consequence we find that the sheaf cohomology group Hq(X,O) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q?1. As another consequence we prove that if f is a C¹-smooth -closed (0,1)-form on the space X=L₁[0,1] of summable functions, then there is a C¹-smooth function u on X with on X.     

Symmetry and-commutativity of topological*-algebras

An advertibly complete locallym-convex (lmc)^*-algebraE is symmetric if and only if each normed (inverse limit) factorE/N , A, ofE is symmetric in the respective Banach factorE , A, ofE. Every locally C^*-algebra is symmetric. If denotes the continuous positive functionals on an lmc^*-algebraE and withL _f ={x E: f(x ^* x) =0}, thenE is, by definition,-commutative if for anyx, y E.-commutativity and commutativity coincide in lmcC ^*-algebras, so that an lmc^*-algebra with a bounded approximate identity is-commutative if and only if its enveloping algebra is commutative. Several standard results for commutative lmc^*-algebras are also obtained in the-commutative case, as for instance, the nonemptiness of the Gel'fand space of a suitable-commutative lmc^*-algebra, the automatic continuity of positive functionals when the algebras involved factor, as well as that the spectral radius is a continuous submultiplicative semi-norm, when the algebras considered are moreover symmetric. An application of the latter result yields a spectral characterization of-commutativity.     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

Minimal area of zero sets in tube domains of C2

Let Ω be a convex domain in C² symmetric with respect to the origin and let f(z, w) run over the class of analytic functions on Ω which vanish at the origin. What is the minimum of the areas of the zero sets Z(f) in Ω? If Ω is a ball or a four-dimensional cube with edges parallel to the axes, the minimum area is attained only if f is a suitable linear function (Lelong-Rutishauser, Katsnelson-Ronkin). In the present paper it is shown that the latter is the case also if Ω is a tube domain T_D= {(z,w)ϵ C²|(x,u)ϵD}. Thus, somewhat surprisingly, the minimum area of a zero set Z(f) through the origin in a convex symmetric tube domain T_D is precisely twice the area of the base D. The special case where D is a circular disc had been treated earlier by Alexander and Osserman.     

On uniformly continuous Nemytskij operators generated by set-valued functions

Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all Hölder functions ${\varphi : I \to C}Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all H?lder functions j: I ? C{\varphi : I \to C} into the set H _β (I, clb(Z)) of all H?lder set-valued functions f: I ? clb(Z){\phi : I \to clb(Z)} and is uniformly continuous, then

F(x,y)=A(x,y) \text+* B(x),       x ? I, y ? CF(x,y)=A(x,y) \stackrel{*}{\text{+}} B(x),\qquad x \in I, y \in C      19. On h-convexity      Sanja Varošanec 《Journal of Mathematical Analysis and Applications》2007,326(1):303-311 We introduce a class of h-convex functions which generalize convex, s-convex, Godunova-Levin functions and P-functions. Namely, the h-convex function is defined as a non-negative function which satisfies f(αx+(1−α)y)?h(α)f(x)+h(1−α)f(y), where h is a non-negative function, α∈(0,1) and x,y∈J. Some properties of h-convex functions are discussed. Also, the Schur-type inequality is given.      20. A decomposition theorem for χ1-convex sets      Marilyn Breen 《Discrete Mathematics》1982,41(2):131-138 A set S in $\text{R}$ is said to be χ-convex if and only if S does not contain a visually independent subset having cardinality χ. It is natural to ask when an χ-convex set may be expressed as a countable union of convex sets. Here it is proved that if S is a closed χ-convex set in the plane and $\text{R}$ has at most finitely many bounded components, then S is a countable union of convex sets. A parallel result holds in $\text{R}$ when S is a closed χ-convex set which contains all triangular regions whose relative boundaries are in S. However, the result fails for arbitrary χ-convex sets, even in the plane.

On h-convexity

We introduce a class of h-convex functions which generalize convex, s-convex, Godunova-Levin functions and P-functions. Namely, the h-convex function is defined as a non-negative function which satisfies f(αx+(1−α)y)?h(α)f(x)+h(1−α)f(y), where h is a non-negative function, α∈(0,1) and x,y∈J. Some properties of h-convex functions are discussed. Also, the Schur-type inequality is given.     

A decomposition theorem for χ1-convex sets

A set S in $\text{R}$ is said to be χ-convex if and only if S does not contain a visually independent subset having cardinality χ. It is natural to ask when an χ-convex set may be expressed as a countable union of convex sets. Here it is proved that if S is a closed χ-convex set in the plane and $\text{R}$ has at most finitely many bounded components, then S is a countable union of convex sets. A parallel result holds in $\text{R}$ when S is a closed χ-convex set which contains all triangular regions whose relative boundaries are in S. However, the result fails for arbitrary χ-convex sets, even in the plane.