Monday, September 15th, 2014 | 4pm-5pm | Burnside 1205 |

Delft University of Technology

Spherical sets avoiding orthogonal pairs of points

Let a_{n} be the supremum of the Lebesgue (surface) measure of I, where I ranges
over all measurable sets of unit vectors in R_{n}
such that no two vectors in I are
orthogonal, and where the surface measure is normalized so that the whole sphere
gets measure 1. The problem of determining a_{n} was first stated in a 1974 note
by H. S. Witsenhausen, where he gave the upper bound of 1/n using a simple
averaging argument. In a 1981 paper by Frankl and Wilson, the authors proved their
well-known theorem and used it to attack this problem; there it was shown that
a_{n} decreases exponentially.We focus on the case n = 3, where we
improve Witsenhausenâ€™s 1/3 upper bound to 0.313. This is a joint work with Oleg Pikhurko.