math.MG — Metric Geometry

An exponential improvement for Borsuk's problem

Contributed by Andriy Prymak

We present a method for improving the classical exponential base $\sqrt{3/2}$ in the upper bound for Borsuk's partition number. The emphasis is on explaining the method rather than optimizing the bound. The problem is reduced to constructing directions which make an acute angle with every member of a prescribed spherical set. These play the same role as the illuminating directions in Schramm's method. The geometric ingredients are an antipodal pairing, a lift by one coordinate, and reflection in a self-dual cone. This reflection folds Euclidean space into the cone. A translation opposite to the horizontal part of its Gaussian mean makes the required supporting inequalities hold with high probability. The Gaussian Poincar\'e inequality controls the mean, and the formula for translating a Gaussian density bounds the measure cost of this construction.