Aleksandrov–Rassias Problem¶

Ask when a mapping between real normed spaces that preserves one prescribed distance must preserve every distance and therefore be an isometry.

Version: v3 · 2026-09-06 · History
Domain-specific #: 1256
Origin domain: mathematics
Subdomain: functional analysis
Aliases: Aleksandrov problem for unit-distance preservers, Rassias problem, Distance-one preserving problem

Core Idea¶

The Aleksandrov–Rassias Problem is a family of rigidity questions about maps between real normed linear spaces. Let \(X\) and \(Y\) be such spaces and let \(f:X o Y\). Suppose \(f\) has the distance-one preserving property: whenever \(\|x-y\|=1\), one has \(\|f(x)-f(y)\|=1\). Under which additional assumptions—such as continuity, surjectivity, dimension, or geometric properties of the range—does this single-distance condition force \(f\) to preserve all distances? The identity is the local-to-global rigidity implication, not one answer detached from its hypotheses.

Scope of Application¶

The Aleksandrov–Rassias Problem is literal when a result or counterexample tests whether preservation of one prescribed distance by a map of normed spaces forces preservation of every distance under explicit hypotheses.

Real normed spaces. The general formulation compares DOPP with global isometry.
Euclidean spaces. Beckman–Quarles-type rigidity provides a benchmark special case.
Hilbert and inner-product spaces. Orthogonality and sphere geometry can strengthen propagation arguments.
Strictly convex ranges. Range geometry may exclude ambiguous midpoint or segment behavior.
Finite-dimensional settings. Dimension-specific geometric constructions can prove or refute rigidity.
Generalized distance sets. Researchers ask which nonzero distances or sets of distances are conservative.
Approximate preservers. Stability branches quantify how local error controls global deviation.
Preserver problems. The template informs broader questions about mappings fixed by limited invariants.

Clarity¶

State scalar field, dimensions, norms, map direction, preserved distance, whether implication is one-way or bidirectional, and every regularity assumption. Define isometry as equality of all pairwise distances and distinguish it from linear isometry or affine isometry. Do not silently infer injectivity unless the assumptions establish it. Separate a theorem for self-maps from a theorem between two spaces and a universal statement from a named class.

Manages Complexity¶

A general map has infinitely many pairwise distances, so testing global isometry directly is an unbounded constraint family. The problem asks whether one level set of the metric can act as a rigidity generator. This converts a global preservation question into a sparse premise plus geometric propagation. The economy is powerful only when the unit-distance graph and norm geometry carry enough structure. In spaces with weak connectivity or flat unit spheres, local constraints can leave freedom.

Abstract Reasoning¶

Fix the real normed spaces and the mapping under study. 2. Normalize and state the prescribed nonzero distance. 3. Write the exact DOPP implication for all qualifying source pairs. 4. List continuity, surjectivity, dimensional, convexity, and direction assumptions independently. 5. Determine which additional distances can be constructed from unit-distance configurations. 6. Use the geometry to propagate preservation to rational or dense distance subsets where justified.

Knowledge Transfer¶

Problem Space is the strict parent by composition and presupposition. The Aleksandrov–Rassias Problem defines admissible objects—normed spaces, maps, and a one-distance constraint—and asks which hypotheses partition the space into rigid and non-rigid cases. The parent contributes variables, constraints, candidate solutions, and boundary exploration. The mathematical residual is DOPP and the sought implication to global isometry.

Relationships to Other Abstractions¶

Current abstraction Aleksandrov–Rassias Problem Domain-specific

Parents (1) — more general patterns this builds on

Aleksandrov–Rassias Problem is a kind of Problem Space Prime

Problem Space is the strict parent by composition/presupposition: the named problem organizes a constrained family of mappings and conditional answer regions.

Hierarchy paths (3) — routes to 2 parentless roots

Aleksandrov–Rassias Problem → Problem Space → Representation → Abstraction

Show alternative paths (2)

Neighborhood in Abstraction Space¶

Aleksandrov–Rassias Problem sits in a sparse region of the domain-specific corpus (77^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Metric Geometry & Approximation (13 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08