Firmly nonexpansive and Kirszbraun-Valentine extensions: a constructive approach via monotone operator theory

Abstract

Utilizing our recent proximal-average based results on the constructive extension of monotone operators, we provide a novel approach to the celebrated Kirszbraun-Valentine Theorem and to the extension of firmly nonexpansive mappings.2000 Mathematics Subject Classification: Primary 46C05, 47H09; Secondary 52A41, 90C25.

Definable versions of theorems by Kirszbraun and Helly

“…is called the proximal average of f and g. This construction (cf. [1,2]) plays a key role in extending monotone set-valued maps in the next section. Proof.…”

“…Remark. In the proof of the result analogous to Proposition 4.12 in [2], appeals to more general results replace our use of the elementary Lemmas 4.9 and 4.10.…”

“…(i) If B = [1,2], then the extension F of f to a function on R m defined as in the proof of Lemma 6.6 is also uniformly continuous. (This is shown for R = R in [24], and the proof given there goes through in general.)…”

“…The proof of Theorem A is based on a recent constructive approach to Kirszbraun's theorem due to Bauschke and Wang [1,2] using the proximal average of convex functions. This is the culmination of a long train of thought (going back at least to Minty [27]) relating Lipschitz maps to monotone set-valued maps.…”

“…This is the culmination of a long train of thought (going back at least to Minty [27]) relating Lipschitz maps to monotone set-valued maps. It is remarkable that the arguments of [1,2] may be transferred in a straightforward way to the setting of definable complete expansions of ordered fields, with the exception of an interesting property of definable families: in general, a family C of closed balls in R n with the finite intersection property may have empty intersection; however (and perhaps, somewhat surprisingly), if R is definably complete and the family C is definable, then C = ∅. More precisely, we have the following result.…”