Quasi‐Isometric Embeddings Between C0(K,X) Spaces Inducing Linear Isometries

ABSTRACT

We establish rigidity of ‐quasi‐isometric embeddings between spaces of the form , where is a uniformly convex Banach space that is uniformly p‐antipodally bounded . By quasi‐isometric rigidity (or qi‐rigidity ), we mean that any such embedding induces a linear isometry from into for some locally compact Hausdorff space .

More precisely, if satisfies for all , with and , then the condition guarantees qi‐rigidity, where is the ‐antipodal constant of and is the associated auxiliary ‐antipodal constant.

This theorem was previously known only for the special case where is a Hilbert space. The present work extends this qi‐rigidity result to a broader class, including all spaces for . This extension also points toward open questions regarding the relationship between qi‐rigidity and the general geometry of ‐uniformly convex spaces.