1 · The essentials
How many smaller pieces does it take to cover a shape? Here, “smaller” means that the greatest distance within each piece is strictly less than the greatest distance within the whole shape. Borsuk’s covering assertion would allow ten such pieces for a set in nine-dimensional space. The manuscript gives a set for which ten pieces are not enough. Its construction represents lines in four-dimensional space by mathematical objects that together form a set in nine-dimensional space. The main counterexample has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine covering a tabletop arrangement with patches, each having a smaller maximum span than the arrangement itself. The result says that, for the particular nine-dimensional arrangement constructed here, ten such patches cannot suffice. The tabletop image does not capture the higher-dimensional geometry.
Why it matters
The result identifies a failure of Borsuk’s covering assertion already in dimension nine. It connects a question about covering sets with a specific construction made from lines in four-dimensional space. The Lean formalization provides formal verification of the main counterexample, rather than only a manuscript account.
2 · Going further
The precise covering question concerns diameter: the greatest distance between points of a set, or its supremum when no greatest distance is attained. In the nine-dimensional case at issue, Borsuk’s assertion asks for a cover by ten sets, each with diameter strictly smaller than that of the original set. The main manuscript gives a compact counterexample, so its failure is explicitly stated for a compact set, not merely for an unrestricted collection of points.
The construction consists of all rank-one orthogonal projectors onto lines in , measured using the Frobenius metric. An orthogonal projector onto a line keeps the component of a vector along that line. For matrices and , the Frobenius distance is , the square root of the sum of the squared differences of their corresponding entries. The source identifies this set as a nine-dimensional counterexample and states that it cannot be covered by ten sets of strictly smaller diameter.
The construction and its covering obstruction are the main idea supplied by the abstract; the source does not describe the argument establishing that obstruction or the details of the nine-dimensional representation. The main result is marked as Lean-formalized. Only one manuscript is supplied, so no relationship to companion manuscripts can be described from this source.
Useful notions
- Distance and diameter
- Covers of sets
- Euclidean spaces
- Lines and orthogonal projections
- Matrices and the Frobenius metric
What was known before / what is new
Before
The source frames the problem as Borsuk’s conjecture, also called Borsuk’s covering assertion. Its nine-dimensional instance would permit a cover by ten sets of strictly smaller diameter. No history of earlier counterexamples or previously known dimension bounds is supplied.
New
The manuscript presents the compact set of rank-one orthogonal projectors on as a counterexample in dimension nine. With the Frobenius metric, this set cannot be covered by ten sets of strictly smaller diameter. The main result has a Lean formalization.
Limits and assumptions
The result concerns the specified compact set with the Frobenius metric and requires every covering set to have strictly smaller diameter. The supplied verification status identifies the main counterexample as Lean-formalized; it does not provide a broader inventory of formalized claims. The source gives neither details of the covering-obstruction argument nor conclusions about lower dimensions, and it does not specify which related questions remain open.
Mini glossary
- Diameter —
- The supremum of the distances between pairs of points in a set.
- Cover —
- A collection of sets whose union contains every point of the set being covered.
- Rank-one orthogonal projector —
- A linear map that sends each vector to its component along a fixed line.
- Frobenius metric —
- The distance between two matrices obtained by taking the square root of the sum of the squared differences of corresponding entries.
- Compact set —
- In Euclidean space, a set that is closed and bounded.
- Lean formalization —
- An encoding of a mathematical statement and its proof for checking by the Lean proof assistant.
Keywords : Borsuk’s conjecture · Diameter · Orthogonal projectors
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)
Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric.
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 nine-dimensional counterexample to Borsuk's covering assertion
23/09/2026The compact set of rank-one orthogonal projectors on ℝ4, with the Frobenius metric, cannot be covered by ten sets of strictly smaller diameter. It therefore gives a counterexample to Borsuk's conjecture in dimension nine.