Proof of the Knop conjecture

In this paper we prove the Knop conjecture asserting that two smooth affine spherical varieties with the same weight monoids are equivariantly isomorphic. We also state and prove a uniqueness property for not necessarily smooth affine spherical varietiesComment: 31 pages, v2,v3 typos fixed, minor mistakes are corrected, references added v4 20 pages, crucial changes made. The proof now uses results of AG/0703543. A new result concerning a uniqueness property of not necessarily smooth affine spherical varieties is added, v6 final version, to appear in Annales de l Institut Fourie