Banach spaces with micro-transitive isometry groups on the sphere are uniformly convex and smooth, form a self-dual class, reduce to separable cases if non-Hilbertian, and hold for L_p only when p=2.
Rolewicz, Metric linear spaces, Mathematics and its Applications 20, PWN-Polish Scientific Publishers, Warsaw
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On Banach spaces whose group of isometries acts micro-transitively on the unit sphere
Banach spaces with micro-transitive isometry groups on the sphere are uniformly convex and smooth, form a self-dual class, reduce to separable cases if non-Hilbertian, and hold for L_p only when p=2.