Petar S. Kenderov

Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections

Two new continuity properties for set-valued mappings are defined which are weaker than lower semicontinuity. One of these properties characterizes when approximate selections exist. A few selection theorems characterized by the other property are established. Some applications are made to set-valued metric projections.

Generic well-posedness of optimization problems in topological spaces

Let X be a completely regular Hausdorff topological space and let C ( X ) (the set of all real-valued bounded and continuous in X functions) be endowed with the sup-norm. Let s X , as usual, denotes the Stone-Cech compactification of X . We give a characterization of those X for which the set contains a dense -subset of C ( X ). These are just the spaces X which contain a dense Cech complete subspace. We call such spaces almost Cech complete. We also prove that X contains a dense completely metrizable subspace, if, and only if, C ( X ) contains a dense -subset of functions which determine Tykhonov well-posed optimization problems over X . For a compact Hausdorff topological space X the latter result was proved by Coban and Kenderov [CK1.CK2]. Relations between the well-posedness and Gâteaux and Frechet differentiability of convex functionals in C ( X ) are investigated. In particular it is shown that the sup-norm in C ( X ) is Frechet differentiable at the points of a dense -subset of C ( X ), if, and only if, the set of isolated points of X is dense in X . Conditions and examples are given when the set of points of Gateaux differentiability of the sup-norm in C ( X ) is a dense and Baire subspace of C ( X ) but does not contain a dense -subset of C ( X ).

Approximation of convex bodies by polytopes

LetC be a convex body ofEd and consider the symmetric difference metric. The distance ofC to its best approximating polytope having at mostn vertices is 0 (1/n2/(d−1)) asn→∞. It is shown that this estimate cannot be improved for anyC of differentiability class two. These results complement analogous theorems for the Hausdorff metric. It is also shown that for both metrics the approximation properties of «most» convex bodies are rather irregular and that ford=2 «most» convex bodies have unique best approximating polygons with respect to both metrics.

Densely defined selections of multivalued mappings

Rather general suficient conditions are found for a given multivalued mapping F: X -* Y to possess an upper semicontinuous and compactvalued selection G which is defined on a dense G,-subset of the domain of F. The case when the selection G is single-valued (and continuous) is also investigated. The results are applied to prove some known as well as new results concerning generic differentiability of convex functions, Lavrentieff type theorem, generic well-posedness of optimization problems and generic nonmultivaluedness of metric projections and antiprojections.

Minimal convex uscos and monotone operators on small sets

We generalize the generic single-valuedness and continuity of monotone operators defined on open subsets of Banach spaces of class (S) and Asplund spaces to monotone operators defined on convex subsets of such spaces which may even fail to have non-support points. This yields differentiability theorems for convex Lipschitzian functions on such sets. From a result about minimal convex uscos which are densely single-valued we obtain generic differentiability results for certain Lipschitzian realvalued functions.

Most of the Optimization Problems have Unique Solution

Let X be a compact metric space and C(X) be the space of all continuous real-valued functions in X. Every pair (A, f), where A belongs to the set 2X of all closed subsets of X and f is from C(X), determines a (constrained) minimization problem: min { f(y): y ∈ A } (find x A at which f attains its minimum over A). Suppose that 2X is endowed with the Hausdorff metric and C(X) is topologized by the usual uniform convergence norm. We prove that there is a dense Gδ-subset G of 2X C(X) such that every minimization problem (A, f) from G has unique solution, i.e. the set { x ∈ A: f(x) = min { f(y) : y ∈ A } } consists of only one point for each pair (A, f) outside some first Baire category subset of 2X×C(X).

Continuity points of quasi-continuous mappings

Abstract It is known that the fragmentability of a topological space X by a metric whose topology contains the topology of X is equivalent to the existence of a winning strategy for one of the players in a special two players “fragmenting game”. In this paper we show that the absence of a winning strategy for the other player is equivalent to each of the following two properties of the space X : for every quasi-continuous mapping f :Z→X , where Z is a complete metric space, there exists a point z 0 ∈Z at which f is continuous; for every quasi-continuous mapping f :Z→X , where Z is an α -favorable space, there exists a dense subset of Z at the points of which f is continuous. In fact, we show that the set of points of continuity of f is of the second Baire category in every non-empty open subset of Z . Using this we derive some results concerning joint continuity of separately continuous functions.

Polygonal approximation of plane convex compacta

Let R2 be the usual two-dimensional plane with the Eucledean norm 1 . I. By CONV we denote the set of all convex compact subsets of R2. The Hausdorff distance between two elements A,, A, of CONV is given by h(A,,A,)=inf{t>O: A,cA,+fB, A,cA,+fB}, where B=(PER2: IPI 3 we denote by POLY, the set of all convex polygons with not more than n vertices. The elements of POLY, will be called n-gons. The ngon A, is said to be a best Hausdorff approximation in POLY, for the set A E CONV if inf{h(A, A): A E POLY,} = h(A, A,). The existence of at least one best Hausdorff approximation for any A E CONV follows from the wellknown Blaschke “selection theorem” asserting that every bounded sequence of n-gons (n fixed) contains a subsequence converging in the Hausdorff metric to some n-gon. In general, as examples like the unit circle or the unit square show, the best approximation is not unique. Nevertheless the “majority” of the elements of CONV have unique best approximation in any POLY,, n > 3. The “majority” here means: with an exception of some first Baire category subset of the locally compact metric space (CONV, h), all convex compact subsets of R2 have unique best approximation in POLY, for every n > 3 (Theorem 3.5). To prove this we give (and use) a necessary condition for A E POLY, to be a best approximation for A E CONV. This condition (Theorem 2.1) coincides with the classical alternating condition in

A generic factorization theorem

Let F : Z → X be a minimal usco map from the Baire space Z into the compact space X . Then a complete metric space P and a minimal usco G : P → X can be constructed so that for every dense G δ -subset P 1 of P there exist a dense G δ Z 1 of Z and a (single-valued) continuous map f : Z 1 → P 1 such that F ( Z )⊂ G ( f ( z )) for every z ∈Z 1 . In particular, if G is single valued on a dense G δ -subset of P , then F is also single-valued on a dense G δ -subset of its domain. The above theorem remains valid if Z is Cech complete space and X is an arbitrary completely regular space. These factorization theorems show that some generalizations of a theorem of Namioka concerning generic single-valuedness and generic continuity of mappings defined in more general spaces can be derived from similar results for mappings with complete metric domains. The theorems can be used also as a tool to establish that certain topological spaces contain dense completely metrizable subspaces.

On the ARG MIN multifunction for lower semicontinuous functions

The epi-topology on the lower semicontinuous functions L(X) on a Hausdorff space X is the restriction of the Fell topology on the closed subsets of X x R to L(X), identifying lower semicontinuous functions with their epigraphs. For each / 6 L(X), let arg min/ be the set of minimizers of /. With respect to the epi-topology, the graph of arg min is a closed subset of L(X) x X if and only if X is locally compact. Moreover, if X is locally compact, then the epi-topology is the weakest topology on L(X) for which the arg min multifunction has closed graph, and the operators /->/Vj and /-t/Aj are continuous for each continuous real function g on X. 1. Introduction. An extended real valued function on a topological space A is called lower semicontinuous (l.s.c.) if for each real a, {x: f(x) > a} is an open subset of A. Equivalently (11), / is lower semicontinuous if its epigraph epi f = {(x, a) : a £ R and a > f(x)} is a closed subset of A x R. In the sequel, we denote the lower semicontinuous functions on A by L(X), and the real valued continuous functions on A by C(X). The fundamental notion of convergence for lower semicontinuous functions proceeds from the identification of such a function