Gevirtz–Omladič–Šemrl Theorem Overview

• The theorem shows that every δ-surjective approximate isometry in Banach spaces can be uniformly approximated by a true surjective linear isometry within a 2δ bound.
• It provides an additive representation for continuous bivariate operations under permutability, expressing them via strictly monotonic functions that mirror affine behavior.
• The work refines the Hyers–Ulam stability framework by delivering explicit quantitative bounds and reduction strategies, extending classical results in functional and metric analysis.

The Gevirtz–Omladič–Šemrl theorem comprises two distinct yet closely related results fundamental to the stability theory of isometries in Banach spaces and to the functional representation theory for permutability equations. These results characterize when approximate isometries and certain bivariate operations, subject to natural algebraic or metric conditions, must be close to linear or additive structures. This theorem forms the cornerstone for several areas, including geometric functional analysis, the theory of functional equations, and the mathematical foundations of empirical laws in physics and measurement theory.

1. Statement and Definitions

Isometric Approximation in Banach Spaces

Let $V$ and $W$ be real Banach spaces with norms $\|\cdot\|_V$ and $\|\cdot\|_W$. For $\delta \ge 0$, a mapping $f\colon V \to W$ is called a $\delta$-isometry if

$$\left| \|f(x) - f(x')\|_W - \|x - x'\|_V \right| \le \delta \quad \text{for all } x, x' \in V.$$

It is $\varepsilon$-surjective if the Hausdorff distance

$$d_H(f(V), W) = \sup_{y \in W} \inf_{x \in V} \|f(x) - y\|_W < \varepsilon.$$

The Mazur–Ulam theorem asserts that any surjective $0$-isometry between Banach spaces is affine. The Gevirtz–Omladič–Šemrl theorem addresses the stability of approximate surjective isometries: For all $\delta \ge 0$, every surjective $\delta$-isometry $f\colon V \to W$ can be uniformly approximated by a true surjective linear isometry $U\colon V \to W$ such that

$$\|f-U\| = \sup_{x \in V} \|f(x) - U(x)\|_W \le 2\delta,$$

where $2$ is the best possible constant (Bogatyi et al., 28 Dec 2025).

Additive-Type Representation for Permutability/Bisymmetry

For real intervals $J, J'$, consider a function $G\colon J \times J' \to J$ that is continuous, strictly increasing in the first variable, strictly monotonic in the second, and satisfies

$$G(G(x, y), z) = G(G(x, z), y) \quad \forall x \in J,\, y, z \in J'.$$

Under these hypotheses, the Gevirtz–Omladič–Šemrl theorem asserts that there exist continuous functions $f\colon J \to \mathbb{R}$ (strictly increasing) and $g\colon J' \to \mathbb{R}$ (strictly monotonic) such that

$G(x, y) = f^{-1}(f(x) + g(y)),$

and in the symmetric case $J' = J$, $G(x, y) = f^{-1}(f(x) + f(y))$ (Falmagne, 2012).

2. Core Theoretical Contributions

Stability of Approximate Isometries

The theorem originated from efforts to resolve the Hyers–Ulam problem for Banach spaces: can an approximate surjective isometry be closely “shadowed” by an exact isometry? J. Gevirtz (1983) established the existence of an affine isometry $U$ with $\|f-U\| \le K\delta$. M. Omladič and P. Šemrl (1995) determined that $K=2$ is best possible. The argument combines metric-structure preservation with affine extension theory.

Recent work generalizes underlying spaces to cardinality-homogeneous metric spaces of equal cardinality and reduces the problem to the classical Banach-space case by constructing bijective approximate isometries and applying the sharp Omladič–Šemrl bound (Bogatyi et al., 28 Dec 2025).

Additive Representations and Permutability

The theorem encapsulates classical results in the representation of bivariate operations via functional equations exhibiting bisymmetry or permutability. If $G$ satisfies the functional equation and the regularity/monotonicity conditions, then $G$ must have the additive-type representation above. This links the structure of the function directly to the abelian group law under monotonicity and mild solvability hypotheses, thereby covering a wide array of scientific and measurement laws ((Falmagne, 2012), Theorem 9).

3. Reduction Strategies and Key Lemmas

Construction of Bijective Approximate Isometries

Reduction from general $\gamma$-surjective $d$-isometries to bijections is achieved via set-theoretic and topological tools:

• Proposition 7.2: For $f\colon X\to Y$ a $\gamma$-surjective $d$-isometry between cardinality-homogeneous metric spaces of equal cardinality, there exists a bijection $\tilde f\colon X\to Y$ that is a $(d+2)$-isometry.
• Proposition 7.6: Extends the conclusion to closed subspaces.
• Cantor–Bernstein–type Construction: Provides bijections matching points within small metric fibers.
• Metric Covering Arguments: Bijections constructed preserve approximate isometry up to explicit distortion bounds (Bogatyi et al., 28 Dec 2025).

Additive Representation via Functional Equations

The bounded Hölder lemma facilitates representation results under finite domains and right-solvability but without global Archimedean hypotheses. Additive representation is constructed via deduction using associativity/commutativity and finiteness properties, allowing for bounded intervals (Falmagne, 2012).

4. Main Illustrative Corollaries and Examples

Quantitative Bounds

The sharp quantitative results include:

• Tabor’s Theorem: Any $\gamma$-surjective $d$-isometry between Banach spaces is $\le 2d+4$ from a surjective affine isometry, improving over earlier bounds ((Bogatyi et al., 28 Dec 2025), Corollary 7.3).
• Dilworth’s Theorem: For mappings onto closed subspaces, an affine isometry is within $2d+12$ ((Bogatyi et al., 28 Dec 2025), Corollary 7.5).

Applications to Scientific and Geometric Laws

Prototypical laws admitting the additive representation include:

• Lorentz–FitzGerald Contraction,
• Beer’s Law of Absorbance,
• Volume of a Cylinder,
• Pythagorean Theorem.

All admit additive representations of the form $G(x, y) = f^{-1}(f(x) + g(y))$ under the theorem’s stated conditions ((Falmagne, 2012), §4).

Example	Formula	Additive Representation
Lorentz–FitzGerald contraction	$L(\ell, v) = \frac{\ell}{\sqrt{1-(v/c)^2}}$	$f(\ell) = \ln \ell$, $g(v) = \frac{1}{2}\ln(1-(v/c)^2)^{-1}$
Beer's law	$I(x, y) = x e^{-c y}$	$f(x) = \ln x$, $g(y) = -cy$
Pythagorean theorem	$P(x, y) = \sqrt{x^2 + y^2}$	$f(x) = x^2$

A key counterexample is the van der Waals equation, which does not satisfy the permutability/bisymmetry requirement (Falmagne, 2012).

5. Broader Context and Generalizations

Extensions to Bounded Domains

Classical additive representation results assume global, Archimedean domains. The 2012 development introduces a finiteness condition replacing infinite divisibility, thus broadening the class of domains to bounded intervals and facilitating applications in measurement theory where only finite steps are possible ((Falmagne, 2012), Lemma 7).

Connections to Functional Stability and Psychophysics

The theorem underpins the stability theory developed by Hyers, Ulam, and others, offering explicit distortion bounds for approximation of measurement rules or physical laws by mathematical group operations. It covers empirical scenarios typical in psychophysical scaling, biomechanics, and relativity, where quantities are naturally bounded (Falmagne, 2012).

6. Impact, Related Results, and Refinements

The Gevirtz–Omladič–Šemrl theorem has become a fundamental tool in the stability of functional equations, metric geometry, and measurement theory. Notable refinements include the sharp constants obtained by Šemrl and Väisälä, and systematic generalizations to non-surjective mappings and arbitrary metric spaces satisfying cardinality homogeneity (Bogatyi et al., 28 Dec 2025). Its methods also underpin reductions of general stability assertions (e.g., Dilworth–Tabor) to the core case of surjective mappings, providing a template for further extensions.

A plausible implication is increased applicability to situations with bounded or countable domains, as the transition from the classical Archimedean axioms to boundedness via finiteness allows the theorem to handle models with limited divisibility or measurement resolution (Falmagne, 2012).