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.
Finet, Uniform convexity properties of norms on a super-reflexive Banach space, Israel J
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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.