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Shift-invariant operators for surface meshes are defined using geometric realizations of the mesh. Then, shift-invariance essentially means isotropy w.r.t. a distance metric. The particular case of the so-defined LSI operators with small support is analyzed in detail, showing a connection to mean value coordinates. The topological Laplacian operator turns out to be the LSI operator of the topological realization of the mesh. More generally, assuming different geometric realizations or metrics allows interpreting various mesh processing techniques as LSI operators.