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We show that the Lusternik-Schnirelmann category of a simple, simplyconnected, compact Lie group G is bounded above by the sum of the relativecategories of certain distinguished conjugacy classes in G corresponding to thevertices of the fundamental alcove for the action of the affine Weyl group onthe Lie algebra of a maximal torus of G.

Distinguished orbits and the L-S category of simply connected compact Lie groups