Nonexpansive Retractions in Hyperconvex Spaces

Abstract

DEDICATED TO GEORGE LEITMANNThis paper is primarily concerned with the study of conditions on a hyperconvex subset D of a hyperconvex metric space M which assure that there exists a nonexpansive retraction R of M\D onto D which has the property that R M\D ⊂ ∂D A related question we take up is, when is such a retraction R proximinal, that is, when does R have the propertyMost of this research was conducted while the first author was visiting the University of Iowa. He thanks the faculty and staff of the Mathematics Department at Iowa for their kind hospitality. Also, the first and third authors acknowledge the support of DGICYT Research Project PB96-1338-C02-01. espínola, kirk, and lópez for each x ∈ M? Among other things, we show that if a subset D of a hyperconvex metricspace M has nonempty interior and is externally hyperconvex relative to M in a very weak sense, then there always exists a nonexpansive retraction of M onto D which maps M\D onto ∂D. We also show that any compact weakly externally hyperconvex subset of a hyperconvex M is a proximinal nonexpansive retract of M.

Introduction to Hyperconvex Spaces

“…We do not plan to show the proof of this result in this text which may be found in [15], however the following theorem is a partial result in that sense, in fact it states a reciprocal of Theorem 4.12.…”

“…Additionally, the externally hyperconvexity of C(x) follows as an easy consequence of Lemma 5.3. Let us see now that C is nonexpansive, for this pick u and v in H. We have to show that C(u) This problem was taken up again in [15] with some improvements of the above theorem. We state these results without proofs, which may be found in [15].…”

Metric fixed point theory on hyperconvex spaces: recent progress

“…The complete characterization of such retracts was given in [20]. To show this characterization we need to introduce a new class of sets: these sets were first introduced in [23] and received the name of weakly externally hyperconvex sets.…”

“…The most it was known about weakly externally hyperconvex spaces and nonexpansive proximinal retractions in [23] is summarized in the following result which was already contained in [21, Section 4].…”

“…Since A is weakly hyperconvex and only one of the above balls is centered outside A, it is enough to check that each two of such balls have nonempty intersection. In a case-bycase check it only remains to study those cases involving the ball centered at r n (v); other cases were already studied in [23]. For these cases it is enough to recall that…”