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Borsuk's conjecture fails in dimension nine
Constructs a compact subset of ℝ 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 ℝ, with the Frobenius metric.
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A nine-dimensional counterexample to Borsuk's covering assertion
Sep 23, 2026·paper.pdfLean
The compact set of rank-one orthogonal projectors on ℝ, 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.