A parametric smooth variational principle and support properties of convex sets and functions

We show a modified version of Georgiev's parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, D(∂h) is pathwise connected and R(∂h) has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then R(∂h) is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets.     

Closed convex sets and their best simultaneous approximation properties with applications

We develop a theory of best simultaneous approximation for closed convex sets in a conditionally complete lattice Banach space X with a strong unit. We study best simultaneous approximation in X by elements of closed convex sets, and give necessary and sufficient conditions for the uniqueness of best simultaneous approximation. We give a characterization of simultaneous pseudo-Chebyshev and quasi-Chebyshev closed convex sets in X. Also, we present various characterizations of best simultaneous approximation of elements by closed convex sets in terms of the extremal points of the closed unit ball B _X* of X*.     

Strong Subdifferentiability of Convex Functionals and Proximinality

Using strong subdifferentiability of convex functionals, we give a new sufficient condition for proximinality of closed subspaces of finite codimension in a Banach space. We apply this result to the Banach space K(l₂) of compact operators on l₂ and we show that a finite codimensional subspace Y of K(l₂) is strongly proximinal if and only if every linear form which vanishes on Y attains its norm.     

On support points and continuous extensions

A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by ${\mathcal{BCC}(X)}A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by BCC(X){\mathcal{BCC}(X)} the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S : BCC(X) ? X{S : \mathcal{BCC}(X) \rightarrow X} such that S(K) is a support point of K for each K ? BCC(X){K \in \mathcal{BCC}(X)}. Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X){\mathcal{BCC}(X)}.     

On smooth, nonlinear surjections of banach spaces

It is shown that (1) every infinite-dimensional Banach space admits aC ¹ Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC ^∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces. Supported by an NSF Postdoctoral Fellowship in Mathematics.     

The strong conical hull intersection property for convex programming

The strong conical hull intersection property (CHIP) is a geometric property of a collection of finitely many closed convex intersecting sets. This basic property, which was introduced by Deutsch et al. in 1997, is one of the central ingredients in the study of constrained interpolation and best approximation. In this paper we establish that the strong CHIP of intersecting sets of constraints is the key characterizing property for optimality and strong duality of convex programming problems. We first show that a sharpened strong CHIP is necessary and sufficient for a complete Lagrange multiplier characterization of optimality for the convex programming model problem where C is a closed convex subset of a Banach space X, S is a closed convex cone which does not necessarily have non-empty interior, Y is a Banach space, is a continuous convex function and g:X→Y is a continuous S-convex function. We also show that the strong CHIP completely characterizes the strong duality for partially finite convex programs, where Y is finite dimensional and g(x)=−Ax+b and S is a polyhedral convex cone. Global sufficient conditions which are strictly weaker than the Slater type conditions are given for the strong CHIP and for the sharpened strong CHIP. The author is grateful to the referees for their constructive comments and valuable suggestions which have contributed to the final preparation of the paper.     

Orthogonally Complemented Subspaces in Banach Spaces

In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.     

A weak Grothendieck compactness principle

The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In this article, an analogue of the Grothendieck compactness principle is considered when the norm topology of a Banach space is replaced by its weak topology. It is shown that every weakly compact subset of a Banach space is contained in the closed convex hull of a weakly null sequence if and only if the Banach space has the Schur property.     

On proximinality of convex sets in superspaces

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.     

Dentability and extreme points in Banach spaces

It is shown that a Banach space E has the Radon-Nikodym property (equivalently, every bounded subset of E is dentable) if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points. Using recent work of Namioka, some analogous results are obtained concerning weak¹ strongly exposed points of weak¹ compact convex subsets of certain dual Banach spaces.     

On typical sets in Banach spaces and a parametric Kuratowski–Ulam theorem

Further investigation is done on a phenomenon studied by Zamfirescu in finite dimensions. Among other results it is proved that for most closed bounded sets A in a separable Banach space Y and most ${u \in A}$ , the union of all rays from u that meet A\{u} (resp. do not meet A\{u}) is dense in Y. An infinite-dimensional extension of a theorem of Wieacker is obtained, viz most compacta in a separable Banach space have smooth closed convex hulls.     

Characterizing continuous functions in terms of separated sets

The following theorem is proved: let f:X →Y and suppose points are closed in Y (that is, singleton sets are closed in Y). Then, f is continous if and only if f ^?1 preserves separated sets. We also give a counter example in the event points are not closed in Y.     

Approximation of fixed points and variational solutions for pseudo-contractive mappings in banach spaces

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Assume that every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive mappings which is also a unique solution to variational inequality problem involving ?-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, convex optimization problems, and split feasibility problems. Our result extends many recent important results.     

Affine Approximation of Lipschitz Functions and Nonlinear Quotients

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient mappings between Banach spaces. It is proved, in particular, that any Banach space which is a uniform quotient of L _p , 1 < p < , is already isomorphic to a linear quotient of L _p . Submitted: June 1998, revised: December 1998.     

Inscribed Balls and Their Centers

A ball of maximal radius inscribed in a convex closed bounded set with a nonempty interior is considered in the class of uniformly convex Banach spaces. It is shown that, under certain conditions, the centers of inscribed balls form a uniformly continuous (as a set function) set-valued mapping in the Hausdorff metric. In a finite-dimensional space of dimension n, the set of centers of balls inscribed in polyhedra with a fixed collection of normals satisfies the Lipschitz condition with respect to sets in the Hausdorff metric. A Lipschitz continuous single-valued selector of the set of centers of balls inscribed in such polyhedra can be found by solving n + 1 linear programming problems.     

On exposed functions in Bernstein spaces

For σ > 0, the Bernstein space {ie427-01} consists of those L ¹(ℝ) functions whose Fourier transforms are supported by [−σ, σ]. Since {ie427-02} is separable and dual to some Banach space, the closed unit ball {ie427-03} of {ie427-04} has sufficiently large sets of both exposed and strongly exposed points: {ie427-05} coincides with the closed convex hull of its strongly exposed points. We investigate some properties of exposed points, construct several examples, and obtain as corollaries relations between the sets of exposed, strongly exposed, weak* exposed, and weak* strongly exposed points of {ie427-06}.     

On convex hulls of compact sets of probability measures with countable supports

E. Michael and I. Namioka proved the following theorem. Let Y be a convex G _δ -subset of a Banach space E such that if K ? Y is a compact space, then its closed (in Y) convex hull is also compact. Then every lower semicontinuous set-valued mapping of a paracompact space X to Y with closed (in Y) convex values has a continuous selection. E. Michael asked the question: Is the assumption that Y is G _δ essential? In this note we give an affirmative answer to this question of Michael.     

Reflexivity and the Fixed-Point Property for Nonexpansive Maps

Connections between reflexivity and the fixed-point property for nonexpansive self-mappings of nonempty, closed, bounded, convex subsets of a Banach space are investigated. In particular, it is shown thatl¹(Γ) for uncountable sets Γ andl^∞cannot even be renormed to have the fixed-point property. As a consequence, if an Orlicz space on a finite measure space that is not purely atomic is endowed with the Orlicz norm, the Orlicz space has the fixed-point property exactly when it is reflexive.     

On linear selections of convex set-valued maps

We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x ₁ + x ₂) ⊂ φ(x ₁) + φ(x ₂). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.     

Lipschitz image of a measure-null set can have a null complement

We give two examples which show that in infinite dimensional Banach spaces the measure-null sets are not preserved by Lipschitz homeomorphisms. There exists a closed setD ? ?₂ which contains a translate of any compact set in the unit ball of ?₂ and a Lipschitz isomorphismF of ?₂ onto ?₂ so thatF(D) is contained in a hyperplane. LetX be a Banach space with an unconditional basis. There exists a Borel setA?X and a Lipschitz isomorphismF ofX onto itself so that the setsX/A andF(A) are both Haar null.