Abstract
If the Radon transform of a compactly supported distribution f ne 0 in mathbb {R}^n is supported on the set of tangent planes to the boundary partial D of a bounded convex domain D, then partial D must be an ellipsoid. As a corollary of this result we get a new proof of a recent theorem of Koldobsky, Merkurjev, and Yaskin, which settled a special case of a conjecture of Arnold that was motivated by a famous lemma of Newton.
Highlights
Define a function f0 in the plane by f0(x) = 1 π1 1 − |x|2 for x = (x1, x2) ∈ R2,|x| < 1, and f0(x) = 0 for |x| > 1
For an arbitrary ellipsoidal domain D ⊂ Rn, n > 2, it is easy to construct examples of distributions f supported in D such that the Radon transform R f is supported on the set of tangent planes to the boundary of D
We show that we may assume that the distribution f is even, f (x) = f (−x), which implies that g = R f is even in ω and p separately
Summary
For an arbitrary ellipsoidal domain D ⊂ Rn, n > 2, it is easy to construct examples of distributions f supported in D such that the Radon transform R f is supported on the set of tangent planes to the boundary of D. If there exists a distribution f = 0 with support in D such that the Radon transform of f is supported on the set of supporting planes for D, ∂ D must be an ellipsoid. 2 we will write down an expression for an arbitrary distribution g on the manifold of hyperplanes that is equal to the Radon transform of some compactly supported distribution and is supported on the submanifold of supporting planes to ∂ D.
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