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We provide a somewhat geometric proof of a rigidity theorem by M. Ledoux andC. Xia concerning complete manifolds with non-negative Ricci curvaturesupporting an Euclidean-type Sobolev inequality with (almost) best Sobolevconstant. Using the same technique we also generalize Ledoux-Xia result tocomplete manifolds with asymptotically non-negative curvature.

An alternative proof of a rigidity theorem for the sharp Sobolev constant