Diameters of homogeneous spaces

Abstract

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 dG(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.