Nonempty Convex Subset Research Articles

On the convergence of von Neumann's alternating projection algorithm for two sets

We give several unifying results, interpretations, and examples regarding the convergence of the von Neumann alternating projection algorithm for two arbitrary closed convex nonempty subsets of a Hilbert space. Our research is formulated within the framework of Fejer monotonicity, convex and set-valued analysis. We also discuss the case of finitely many sets.

An application of KKM‐map principle

The following theorem is proved and several fixed point theorems and coincidence theorems are derived as corollaries. Let C be a nonempty convex subset of a normed linear space X, f : C → X a continuous function, g : C → C continuous, onto and almost quasi‐convex. Assume that C has a nonempty compact convex subset D such that the set urn:x-wiley:01611712:media:ijmm571891:ijmm571891-math-0001 is compact.Then there is a point y0 ∈ C such that ‖g(y0) − f(y0)‖ = d(f(y0), C).

The rate of weak convergence of convex type positive finite measures

Let μ be a positive finite measure of mass m of the non-empty convex subset M of the real normed vector space ( V, ∥ · ∥). For a fixed x 0 ϵ M and τ( x) = ∥ x − x 0∥, let the probability measure ϱ = m −1 μ ∘ τ −1. Assume that the corresponding ϱ distribution function fulfills certain convexity conditions. By the use of convex moment methods best upper bounds of ¦∫ M f dμ − f(x 0)¦ are obtained for f an integrable real valued function on M and a given power moment of μ. These lead to sharp inequalities, i.e., attainable inequalities involving the first modulus of continuity of f. The established estimates improve the corresponding ones in the literature. These have wide applications to concave positive linear operators typically arising from well-known probabilistic distributions.

Convex Lp approximation

In [7] Karlovitz developed an algorithm for finding best Lp approximations from certain finite-dimensional subspaces for p an even integer. It will be shown that this algorithm converges for 2 < p < co when the approximating subspace is replaced by certain closed convex subsets of a linitedimensional subspace. Furthermore, the restrictions which must be placed on the functions involved to ensure convergence are weakened. We shall consider the problem of approximating 0 by elements of a closed nonempty convex subset K which is contained in a finite-dimensional subspace and which does not contain 0. This is seen to be equivalent to the general problem of approximating a function f by elements of a closed convex subset G, with f 65 G and G contained in a finite-dimensional subspace, by simply translating all functions involved by -f. That is,

Nearest points and support points in Banach spaces

DEFINITION. Let Y be a family of closed convex subsets of a Banach space X. For a real number, (Y, we shall say X possesses property D(or, 9) if for each ,4 E 9 and each ICE X there is a pomt u E X such that x E u + aPA( (We shall use the convention that u + o = o .) Nashed, in the paper cited above, showed that every Hilbert space possesses D( 1, YO) where Y0 is the famdy of all closed nonempty convex subsets of the space. Our investigation is divided into three sections which consider, respectively, the cases when (Y > 1, 01 = 1, and ar < 1. In each case we obtain a characterization of reflexivity (Theorem 1, Theorem 2, and

Closed subgroups of lattice-ordered permutation groups

Let G G be an l l -subgroup of the lattice-ordered group A ( Ω ) A(\Omega ) of order-preserving permutations of a chain Ω \Omega ; and in this abstract, assume for convenience that G G is transitive. Let Ω ¯ \bar \Omega denote the completion by Dedekind cuts of Ω \Omega . The stabilizer subgroups G ω ¯ = { g ϵ G | ω ¯ g = ω ¯ } , ω ¯ ϵ Ω ¯ {G_{\bar \omega }} = \{ g \epsilon G|\bar \omega g = \bar \omega \} ,\bar \omega \epsilon \bar \Omega , will be used to characterize certain subgroups of G G which are closed (under arbitrary suprema which exist in G G ). If Δ \Delta is an o o -block of G G (a nonempty convex subset such that for any g ϵ G g \epsilon G , either Δ g = Δ \Delta g = \Delta or Δ g ∩ Δ \Delta g \cap \Delta is empty), and if ω ¯ = sup Δ , G Δ \bar \omega = \sup \Delta ,{G_\Delta } will denote { g ϵ G | Δ g = Δ } = G ω ¯ \{ g \epsilon G|\Delta g = \Delta \} = {G_{\bar \omega }} ; and the o o -block system Δ ~ \tilde \Delta consisting of the translates Δ g \Delta g of Δ \Delta will be called closed if G Δ {G_\Delta } is closed. When the collection of o o -block systems is totally ordered (by inclusion, viewing the systems as congruences), there is a smallest closed system C \mathcal {C} , and all systems above C \mathcal {C} are closed. C \mathcal {C} is the trivial system (of singletons) iff G G is complete (in A ( Ω ) A(\Omega ) ). G ω ¯ {G_{\bar \omega }} is closed iff ω ¯ \bar \omega is a cut in C \mathcal {C} i.e., ω ¯ \bar \omega is not in the interior of any Δ ϵ C \Delta \epsilon \mathcal {C} . Every closed convex l l -subgroup of G G is an inter-section of stabilizers of cuts in C \mathcal {C} . Every closed prime subgroup ≠ G \ne G is either a stabilizer of a cut in C \mathcal {C} , or else is minimal and is the intersection of a tower of such stabilizers. L ( C ) = ∩ { G Δ | Δ ϵ C } L(\mathcal {C}) = \cap \{ {G_\Delta }|\Delta \epsilon \mathcal {C}\} is the distributive radical of G G , so that G G acts faithfully (and completely) on C \mathcal {C} iff G G is completely distributive. Every closed l l -ideal of G G is L ( D ) L(\mathcal {D}) for some system D \mathcal {D} . A group G G in which every nontrivial o o -block supports some 1 ≠ g ϵ G 1 \ne g \epsilon G (e.g., a generalized ordered wreath product) fails to be complete iff G G has a smallest nontrivial system Δ ~ \tilde \Delta and the restriction G Δ | Δ {G_\Delta }|\Delta is o o - 2 2 -transitive and lacks elements ≠ 1 \ne 1 of bounded support. These results about permutation groups are used to show that if H H is an abstract l l -group having a representing subgroup, its closed l l -ideals form a tower under inclusion; and that if { K λ } \{ {K_\lambda }\} is a Holland kernel of a completely distributive abstract l l -group H H , then so is the set of closures { K λ ∗ } \{ K_\lambda ^ \ast \} , so that if H H has a transitive representation as a permutation group, it has a complete transitive representation.

Maximal separation theorems for convex sets

Abstract : Most previous applications of separation theorems for convex sets have depended on rather crude separation results. Recent developments, however, indicate a need for more refined theorems, particularly for ones involving stronger types of separation. The stronger types of separation considered in this study include nice, open, closed, strict, and strong. The attempt to obtain separation theorems under minimal hypotheses suggests a search for maximal theorems, since each such theorem is, in a sense, a best possible result. Eighteen maximal theorems for disjoint nonempty convex subsets are derived, involving open, nice, strong and strict separation. (Author)