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 $\beta \approx .12$ (independent of $G$) such that for any closed subgroup $H \subsetneq G$, the diameter of the quotient $G/H$ (in the induced metric) is $\geq \beta$.

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Published 1 January 2003