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A Probabilistic Generalization of the Mazur-Ulam Theorem

Justinas Zaliaduonis, Sergios Gatidis

TL;DR

This work generalizes the Mazur-Ulam rigidity phenomenon to settings where distance preservation holds only almost everywhere with respect to a probability measure $\mu$ with full-dimensional support. The authors develop a measure-theoretic and affine-geometric framework, proving that a $\mu$-a.e. isometry $h$ must coincide a.e. with a global Euclidean isometry $H(x)=Qx+b$ with $Q\in O(d)$. The key steps include selecting $d+1$ affinely independent points in a full-measure set, extending the induced finite isometry to a global one, and using a full-measure intersection argument to conclude $h=H$ a.e. The result bridges affine geometry and measure theory, providing a robust rigidity principle applicable to probabilistic or noisy data contexts and underscoring that almost-everywhere distance preservation implies strong global structure.

Abstract

The classical Mazur-Ulam theorem establishes that every surjective isometry between normed real vector spaces is an affine transformation. In various applied mathematical settings, however, one encounters maps that preserve distances not pointwise, but almost everywhere with respect to a probability measure. This paper provides a rigorous generalization of the Mazur-Ulam theorem to probability spaces. We prove that if a measurable map on a subset of Rd preserves distances almost everywhere with respect to a measure with full-dimensional support, it coincides almost everywhere with a global Euclidean isometry, defined as an orthogonal transformation followed by a translation.

A Probabilistic Generalization of the Mazur-Ulam Theorem

TL;DR

This work generalizes the Mazur-Ulam rigidity phenomenon to settings where distance preservation holds only almost everywhere with respect to a probability measure with full-dimensional support. The authors develop a measure-theoretic and affine-geometric framework, proving that a -a.e. isometry must coincide a.e. with a global Euclidean isometry with . The key steps include selecting affinely independent points in a full-measure set, extending the induced finite isometry to a global one, and using a full-measure intersection argument to conclude a.e. The result bridges affine geometry and measure theory, providing a robust rigidity principle applicable to probabilistic or noisy data contexts and underscoring that almost-everywhere distance preservation implies strong global structure.

Abstract

The classical Mazur-Ulam theorem establishes that every surjective isometry between normed real vector spaces is an affine transformation. In various applied mathematical settings, however, one encounters maps that preserve distances not pointwise, but almost everywhere with respect to a probability measure. This paper provides a rigorous generalization of the Mazur-Ulam theorem to probability spaces. We prove that if a measurable map on a subset of Rd preserves distances almost everywhere with respect to a measure with full-dimensional support, it coincides almost everywhere with a global Euclidean isometry, defined as an orthogonal transformation followed by a translation.
Paper Structure (7 sections, 3 theorems, 15 equations)

This paper contains 7 sections, 3 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Definitions
  3. Measure Theoretic Setup
  4. Affine Geometry
  5. Rigidity of Finite Isometries
  6. Main Result: Probabilistic Mazur-Ulam Theorem
  7. Conclusion

Key Result

Theorem 1.1

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$. If $f: V \to W$ is a surjective isometry, then $f$ is an affine map. That is, $f(x) = T(x) + b$ where $T$ is a linear map and $b \in W$ is a translation vector.

Theorems & Definitions (9)

  • Theorem 1.1: Mazur-Ulam
  • Definition 2.1: Full-Dimensional Support
  • Definition 2.2: Isometry Almost Everywhere
  • Definition 2.3: Affine Independence
  • Definition 2.4: Affine Hull
  • Lemma 3.1: Extension of Finite Isometry
  • proof
  • Theorem 4.1: Probabilistic Mazur-Ulam
  • proof