Borsuk number in four dimensions is at most seven

Abstract

We prove that every bounded set of diameter one in four-dimensional Euclidean space can be covered by seven balls of diameter strictly smaller than $0.9999$. In particular, the Borsuk number satisfies $b_4\le7$. The proof combines geometric arguments based on contact points of the circumball with an exact finite computation. For small and large circumradius, the covering centres are constructed analytically. In the remaining range, a suitable triangle of contact points reduces the problem to a four-parameter family of ball polytopes. The parameter cube is subdivided into 23494 boxes. For each parameter box, candidate centres are found by nonlinear optimization, and the resulting covering is verified using exact rational interval arithmetic and paraboloid lifting.