If K and L are convex bodies whose radial functions have the same distribution function, i.e. equi-measurable, what can be said? In particular, does this mean that the two radial functions(of the sphere) are equal up to composition with some measure-preserving transformation? If so, does the convexity of the two bodies imply any regularity of this measure-preserving transformation of the sphere? If possible, how much regularity must the bodies possess to force this transformation to be a rotation or orthonormal linear transformation of the sphere?
Pure math is always ahead of the game because we pave the roads and identify the dead ends for applied sciences. Sometimes we pave the roads so far ahead of technology that our results get lost or forgotten. I'm sure there are answers to many applied questions hidden away in decades old pure math publications but the applied scientists don't have the vocabulary to even do a useful search for this information. There needs to be more communication from both sides.
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