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Porous sets for mutually nearest points in Banach spaces

100%

Li C. , Myjak J.

Opuscula Mathematica

2008

tom Vol. 28, no. 1

73-82

Let B(X) denote the family of all nonempty closed bounded subsets of a real Banach space X, endowed with the Hausdorff metric. For E, F ∈ B (X) we set [formula]. Let D denote the closure (under the maximum distance) of the set of all (E, F) ∈ B (X) x B (X) such that λE,F > 0. It is proved that the set of all (E, F) ∈ D for which the minimization problem [formula] fails to be well posed in a σ-porous subset of D.

On [sigma]-porous and [Phi]-angle-small sets in metric spaces

100%

Rolewicz S.

Control and Cybernetics

2003

tom Vol. 32, no 3

671-681

It is shown that in metric spaces each [alpha, phi)-meagre set A is uniformly very porous and its index of uniform v-porosity is not smaller than [k-alpha/3k+alpha] provided that [phi] is a strictly k-monotone family of Lipschitz functions and [alpha] < k. The paper contains also conditions implying that a k-monotone family of Lipschitz functions is strictly k-monotone.

Exceptional sets for universally polygonally approximable functions

63%

Evans M. J. , Humke P. D.

Journal of Applied Analysis

2001

tom Vol. 7, nr 2

175-190

In [9], the present authors and Richard O'Malley showed that in order for a function be universally polygonally approximate it is necessary that for each ε > 0, the set of points of non-quasicontinuity be σ - (1 - ε ) symmetrically porous. The question as to whether that condition is sufficient or not was left open. Here we prove that if a set, E = U∞n=1 En, such that each Ei is closed and 1-symmetrically porous, then there is a universally polygonally approximable function, f, whose set of points of non-quasicontinuity is precisely E. Although it is tempting to call this a partial converse to our earlier theorem it might be more since it is not known if these two notions of symmetric porosity differ in the class of F? sets.

Porosity and the bounded linear regularity property

63%

Reich S. , Zaslavski A. J.

Journal of Applied Analysis

2014

tom Vol. 20, nr 1

1--6

H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.