Diameters of Homogeneous Spaces

Let G be a compact connected Lie group with trivial center. Using the action of G on its Lie algebra, we define an operator norm | | G which induces a bi-invariant metric d G (x,y) = | Ad(yx -1 )| G on G . We prove the existence of a constant Β = .12 (independent of G ) such that for any closed subgroup H ⊆ G , the diameter of the quotient G/H (in the induced metric) is ≥ Β .