|Monday, September 15th, 2014||4pm-5pm||Burnside 1205|
Let an be the supremum of the Lebesgue (surface) measure of I, where I ranges over all measurable sets of unit vectors in Rn 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 an 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 an decreases exponentially.We focus on the case n = 3, where we improve Witsenhausen’s 1/3 upper bound to 0.313. This is a joint work with Oleg Pikhurko.