The "size" can mean the number of elements or the diameter of the set of smooth structures. Y. Shikata defined a distance function on it and proved that it is a distance. He then used it to prove that a compact topological metric manifold can be smoothable if and only if the "size" (which is a little complicated) of the manifold is equal to zero. Moreover, he used it to prove a pinched differential sphere theorem. But the pinching constent is very close to one and it depends on the metric which is induced from the Euclidean metric on the sphere. Can we use it to do other things? For example, can we relate it to the moduli space of positive (or negative...) curvature metrics? Or, has there been any progress on this topic?
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