The Four-Dimensional Borsuk Number is Five or Six

Can every set of diameter one in four dimensions be split into at most five parts of smaller diameter? Three parts suffice in the plane and four in three-dimensional space. In four dimensions five parts are sometimes necessary. We prove that six always suffice, improving the previous upper bound of seven. For small circumradius we use six cones. For large circumradius five balls meet every diametral pair of a compact set. At intermediate radii we bound the diameter of each part by the diameter of a rational polytope.