Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
DEFINABLE VERSIONS OF THEOREMS BY KIRSZBRAUN MATTHIAS ASCHENBRENNER AND ANDREAS FISCHER
 

Summary: DEFINABLE VERSIONS OF THEOREMS BY KIRSZBRAUN
AND HELLY
MATTHIAS ASCHENBRENNER AND ANDREAS FISCHER
Abstract. Kirszbraun's Theorem states that every Lipschitz map S Rn,
where S Rm, has an extension to a Lipschitz map Rm Rn with the
same Lipschitz constant. Its proof relies on Helly's Theorem: every family
of compact subsets of Rn, having the property that each of its subfamilies
consisting of at most n + 1 sets share a common point, has a non-empty
intersection. We prove versions of these theorems valid for definable maps and
sets in arbitrary definably complete expansions of ordered fields.
Introduction
Let L be a non-negative real number and let f : S Rn
, S Rm
, be an L-Lipschitz
map, i.e., ||f(x) - f(y)|| L ||x - y|| for all x, y S. It was noted by McShane
and Whitney independently (1934) that if n = 1, then f extends to an L-Lipschitz
function Rm
R. This immediately implies that for general n, there always exists
a

  

Source: Aschenbrenner, Matthias - Department of Mathematics, University of California at Los Angeles

 

Collections: Mathematics