Let R be a region on a hypersphere. Each point A of the hypersphere is associated with a vector pointing to A and with origin at the centre of the hypersphere. So let me identify each point with a vector. I have the constraint that every pair of vectors in R is not orthogonal. What is the maximal region R satisfying this constraint? I guess that this region, up to rotations, is given by two opposite hypercones with 90 degrees angular aperture. This is intuitive, but the proof is not so simple to me.