This is an announcement for the paper "The approximate fixed point property in product spaces" by Ulrich Kohlenbach and Laurentiu Leustean. Abstract: In this paper we generalize to unbounded convex subsets C of hyperbolic spaces results obtained by W.A. Kirk and R. Espinola on approximate fixed points of nonexpansive mappings in product spaces $(C\times M)_\infty$, where M is a metric space and C is a nonempty, convex, closed and bounded subset of a normed or a CAT(0)-space. We extend the results further, to families $(C_u)_{u\in M}$ of unbounded convex subsets of a hyperbolic space. The key ingredient in obtaining these generalizations is a uniform quantitative version of a theorem due to Borwein, Reich and Shafrir, obtained by the authors in a previous paper using techniques from mathematical logic. Inspired by that, we introduce in the last section the notion of uniform approximate fixed point property for sets C and classes of self-mappings of C. The paper ends with an open problem. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 47H10, 47H09 (Primary) 03F10 (Secondary) Remarks: 15 pages The source file(s), AFPP.tex: 38299 bytes, is(are) stored in gzipped form as 0510563.gz with size 10kb. The corresponding postcript file has gzipped size 62kb. Submitted from: leustean@mathematik.tu-darmstadt.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510563 or http://arXiv.org/abs/math.FA/0510563 or by email in unzipped form by transmitting an empty message with subject line uget 0510563 or in gzipped form by using subject line get 0510563 to: math@arXiv.org.