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Extension of Lipschitz maps definable in Hensel minimal structures

Krzysztof Jan Nowak

TL;DR

The paper proves a Kirszbraun-type theorem for extending definable 1-Lipschitz maps from subsets of K^n to K^n in 1-h-minimal, equicharacteristic zero non-Archimedean fields. It develops a robust framework based on definable open cell packages with skeletons and risometries, and employs a double induction on ambient and domain dimensions. Key techniques include RV-parameter compatible cell decomposition, the LE-property, and the valuative Jacobian property, which together enable both ordinary and uniform (parameterized) extensions. The results extend the definable Lipschitz extension theory to a broad non-locally-compact, non-Archimedean setting and generalize prior p-adic and o-minimal analogues with definable Skolem-function considerations.

Abstract

In this paper, we establish a theorem on extension of Lipschitz maps $f$ definable in Hensel minimal, non-trivially valued fields $K$ of equicharacteristic zero. 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.

Extension of Lipschitz maps definable in Hensel minimal structures

TL;DR

The paper proves a Kirszbraun-type theorem for extending definable 1-Lipschitz maps from subsets of K^n to K^n in 1-h-minimal, equicharacteristic zero non-Archimedean fields. It develops a robust framework based on definable open cell packages with skeletons and risometries, and employs a double induction on ambient and domain dimensions. Key techniques include RV-parameter compatible cell decomposition, the LE-property, and the valuative Jacobian property, which together enable both ordinary and uniform (parameterized) extensions. The results extend the definable Lipschitz extension theory to a broad non-locally-compact, non-Archimedean setting and generalize prior p-adic and o-minimal analogues with definable Skolem-function considerations.

Abstract

In this paper, we establish a theorem on extension of Lipschitz maps definable in Hensel minimal, non-trivially valued fields of equicharacteristic zero. 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 . 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.
Paper Structure (6 sections, 11 theorems, 208 equations)

This paper contains 6 sections, 11 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. When algebraic Skolem functions are available
  4. Lipschitz extension on the affine line
  5. Lipschitz extension on the affine plane
  6. Lipschitz extension for higher dimensions

Key Result

Theorem 1.1

Let $f: A \to K^m$ be a $0$-definable 1-Lipschitz map on a (possibly non-closed) subset $A \subset K^n$ of dimension $k$. I. Suppose the value group $|K|$ has no minimal element among the elements $>1$. Then, for any $\epsilon \in |K|$, $\epsilon > 1$, $f$ extends to a $0$-definable $\epsilon$-Lipsc

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 18 more