On Aumann's Theorem that the sphere does not admit a mean
  We prove that the circle S_1 does not have a 2-mean, i.e., S_1 times S_1 cannot have a retraction r onto its diagonal with r(x,y) = r(y,x), whenever x,y in S_1. Our proof is combinatorial and topological rather than analytical. 
F. Javier Trigos-Arrieta, Marian Turzanski