Title: Maximum measures of spherical sets avoiding a prescribed set of angles
Abstract: 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 combinatorial averaging argument. In this talk, we focus on the case n=3,
where we improve Witsenhausen’s 1/3 upper bound to 0.313. The proof involves some
basic harmonic analysis and infinite-dimensional linear programming.
It turns out that the above supremum is actually a maximum. Time permitting, we
shall discuss the following more general question: For an arbitrary subset X of
[-1,1], call a subset I of the unit sphere in R^n X-independent if no two vectors
in I make an inner product lying in X, and let a_n(X) be the supremum of the
measures of all X-independent sets. Is a_n(X) a maximum? The answer is Yes for all
X when n>=3, and No in general for n=2. This is joint work with Oleg Pikhurko.