1 · The essentials
Can a set with a simple pattern of distances always be represented using finitely many coordinates without stretching or shrinking those distances too unevenly? The main manuscript gives a negative answer. In every infinite-dimensional real Banach space—a setting for measuring the sizes of vectors—it finds a compact set, meaning roughly that the set has no missing limit points and cannot spread out indefinitely. The set also has a uniform local simplicity: any portion inside a ball can be covered by a bounded number of balls of half the radius. Yet no representation in any finite-dimensional normed space can keep all distances between fixed positive multiples of their original values. The bound describing that local simplicity can be chosen independently of the surrounding space. The main result has a Lean formalization and answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a map whose every neighborhood can be covered by a fixed number of smaller map tiles, at every zoom level. That orderly tiling rule still does not guarantee that the map can be placed in a space with finitely many coordinates while keeping every pairwise distance within fixed stretch-and-shrink limits.
Why it matters
The result separates two kinds of geometric simplicity: having uniformly controlled coverings at every scale and admitting a finite-dimensional representation with controlled distortion of distances. The manuscript answers the Lang–Plaut problem negatively even under compactness and even inside Hilbert space. Its universal covering bound means this obstruction is not dependent on choosing a different bound for each ambient Banach space.
2 · Going further
The precise statement is that every infinite-dimensional real Banach space contains a compact doubling subset with no bi-Lipschitz embedding into any finite-dimensional real normed space. Doubling means that there is a fixed bound such that each ball in can be covered by at most balls of half its radius. Here has a universal bound independent of .
A bi-Lipschitz embedding would be a map for which constants satisfy for all . The result rules out such a map for every finite-dimensional real normed target , not merely for a particular target dimension or a particular choice of distortion bounds.
The source identifies this as a negative answer to the Lang–Plaut problem, including the case of compact subsets of Hilbert space. The abstract describes the counterexample through its properties but supplies no construction or argument, so an account of the proof strategy cannot be extracted from the supplied text. Only the main manuscript is supplied; it is marked Lean-formalized, and no companion manuscripts or their relationships are described.
Useful notions
- Distances and metric balls
- Real normed spaces and Banach spaces
- Compactness
- Doubling coverings
- Bi-Lipschitz embeddings
What was known before / what is new
Before
The source identifies the Lang–Plaut problem as the question answered negatively by this result. It supplies no earlier partial results or historical account, so no more detailed description of the prior state is available here.
New
The main result provides a compact doubling counterexample in every infinite-dimensional real Banach space. The doubling constant has a universal bound, while the set admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The source explicitly includes compact subsets of Hilbert space in this negative conclusion.
Limits and assumptions
The ambient-space hypothesis is that the space is real, infinite-dimensional, and Banach. The conclusion concerns the existence of a particular compact doubling subset, not every subset, and excludes bi-Lipschitz embeddings rather than arbitrary maps. The main result has a Lean formalization; the supplied text gives no finer account of its formalization scope or construction details. It states that the Lang–Plaut problem is answered negatively but identifies no remaining open questions.
Mini glossary
- Banach space —
- A Banach space is a normed vector space in which every Cauchy sequence converges.
- Compact set —
- In a metric space, a set is compact if every sequence in it has a subsequence converging to a point of the set.
- Doubling set —
- A metric set is doubling if every ball can be covered by a uniformly bounded number of balls of half its radius.
- Bi-Lipschitz embedding —
- A bi-Lipschitz embedding is a map that keeps every distance between fixed positive lower and upper multiples of the original distance.
- Hilbert space —
- A Hilbert space is a complete inner-product space, with distances given by its associated norm.
Keywords : Doubling sets · Bi-Lipschitz embeddings · Dimension reduction
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding
25/09/2026Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.