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Porosity and the bounded linear regularity property

Reich S., Zaslavski A. J.

Journal of Applied Analysis

2014

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.

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

Rolewicz S.

Control and Cybernetics

2003

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.