OpenAI research collection · 1 manuscript · Source checked 2026-10-19
Constructs a compact subset of ℝ^(9) 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^(4), with the Frobenius metric.
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The compact set of rank-one orthogonal projectors on ℝ^(4), 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.
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