 
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 nonempty
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 nonnegative real number and let f : S Rn
, S Rm
, be an LLipschitz
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 LLipschitz
function Rm
R. This immediately implies that for general n, there always exists
a
