Extreme Punkte in der Einheitskugel des Vektorraumes der trigonometrischen Prolynome.
Fiber orders and compact spaces of uncountable weight
We study an order relation on the fibers of a continuous map and its application to the study of the structure of compact spaces of uncountable weight.
Fixed points, intersection theorems, variational inequalities, and equilibrium theorems.
Fix-finite approximation property in normed vector spaces.
General methods of constructing equivalent minimal pairs not unique up to translation.
In this paper we give general methods of construction of various equivalent minimal pairs of compact convex sets that are not translates of one another.
Invariants of pairs of compact convex sets.
Konvexitätstreue und Linearität von Abbildungen.
Locally Countable Plump Tilings are Flat.
Maximal elements and equilibria of generalized games for -majorized and condensing correspondences.
Metrization of Compact Convex Sets.
m-families and Common Transversals
Minimal pairs of bounded closed convex sets
The existence of a minimal element in every equivalence class of pairs of bounded closed convex sets in a reflexive locally convex topological vector space is proved. An example of a non-reflexive Banach space with an equivalence class containing no minimal element is presented.
Minimal pairs of compact convex sets
Pairs of compact convex sets naturally arise in quasidifferential calculus as sub- and superdifferentials of a quasidifferentiable function (see Dem86). Since the sub- and superdifferentials are not uniquely determined, minimal representations are of special importance. In this paper we give a survey on some recent results on minimal pairs of closed bounded convex sets in a topological vector space (see PALURB). Particular attention is paid to the problem of characterizing minimal representatives...
Minimal pairs representing selections of four linear functions in .
Minimality in asymmetry classes
We examine minimality in asymmetry classes of convex compact sets with respect to inclusion. We prove that each class has a minimal element. Moreover, we show there is a connection between asymmetry classes and the Rådström-Hörmander lattice. This is used to present an alternative solution to the problem of minimality posed by G. Ewald and G. C. Shephard in [4].
Minimax equalities by reconstruction of polytopes.
Minimax inequality of Ky Fan type in -spaces with applications.
More on exposed points and extremal points of convex sets in and Hilbert space
Let be a separable real Hilbert space, with , and let be convex and closed in . Let be a collection of linear -subspaces of . A point is called exposed by if there is a so that . We show that, under some natural conditions, can be reconstituted as the convex hull of the closure of all its exposed by points whenever is dense and . In addition, we discuss the question when the set of exposed by some points forms a -set.
Note on Bessaga-Klee classification
We collect several variants of the proof of the third case of the Bessaga-Klee relative classification of closed convex bodies in topological vector spaces. We were motivated by the fact that we have not found anywhere in the literature a complete correct proof. In particular, we point out an error in the proof given in the book of C. Bessaga and A. Pełczyński (1975). We further provide a simplified version of T. Dobrowolski's proof of the smooth classification of smooth convex bodies in Banach...
On closed sets with convex projections in Hilbert space
Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that is precisely the intersection of all k-imitations...