Extension of Lipschitz maps definable in Hensel minimal structures

TL;DR

This paper proves a version of Kirszbraun's extension theorem for Lipschitz maps definable in Hensel minimal fields, using a novel induction approach involving definable open cell packages and risometries.

Contribution

It introduces a definable extension theorem for Lipschitz maps in Hensel minimal structures, expanding non-Archimedean geometric analysis.

Findings

Established a non-Archimedean Kirszbraun-type extension theorem

Developed a double induction method based on dimension

Introduced the concepts of definable open cell packages and risometries

Abstract

In this paper, we establish a theorem on extension of Lipschitz maps $f$ definable in Hensel minimal fields $K$. This may be regarded as a definable, non-Archimedean, non-locally compact version of Kirszbraun's extension theorem. We proceed with double induction with respect to the dimensions of the ambient space and of the domain of $f$. To this end, we introduce the concept of a definable open cell package with a skeleton which, along with the concept of a risometry, plays a key role in our induction procedure.