The smallest dimension of a counterexample to Borsuk's conjecture: failure in dimension nine

Theorem 1.1: the compact set of rank-one orthogonal projectors $uu^T$, $u\in\mathbb R^4$ a unit vector, with the Frobenius metric, lies in the nine-dimensional affine space of trace-one symmetric $4\times4$ matrices, has diameter $\sqrt2$ and cannot be covered by ten sets of smaller diameter, so Borsuk's assertion fails in dimension 9 (the record was 63). A corollary extends this to compact counterexamples in every dimension $d\ge9$. Not shown: the smallest failing dimension (dimensions 4 to 8 remain open) or the exact Borsuk number of the set. The witness is a continuum, not a finite point set.

Theorem 1.1: the compact set of rank-one orthogonal projectors $uu^T$, $u\in\mathbb R^4$ a unit vector, with the Frobenius metric, lies in the nine-dimensional affine space of trace-one symmetric $4\times4$ matrices, has diameter $\sqrt2$ and cannot be covered by ten sets of smaller diameter, so Borsuk's assertion fails in dimension 9 (the record was 63). A corollary extends this to compact counterexamples in every dimension $d\ge9$. Not shown: the smallest failing dimension (dimensions 4 to 8 remain open) or the exact Borsuk number of the set. The witness is a continuum, not a finite point set.