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Radon transforms supported in hypersurfaces and a conjecture by Arnold

Banff International Research Station for Mathematical Innovation and Discovery

A famous lemma in Newton's Principia says that the area of a segment of a bounded convex domain in the plane cannot depend algebraically on the parameters of the line that defines the segment. Vassiliev extended Newton's lemma to bounded convex domains in arbitrary even dimensions. In odd dimensions the volume cut out from an ellipsoid by a hyperplane depends not only algebraically but polynomially on the position of the hyperplane. Arnold conjectured in 1987 that ellipsoids in odd dimensions are the only cases in which the volume function in question is algebraic. The special case when the volume function is assumed to be polynomial was settled recently by Koldobsky, Merkurjev, and Yaskin. Motivated by a totally different problem I tried to construct a compactly supported distribution $f\ne 0$ whose Radon transform is supported in the set of tangent planes to the boundary surface $\partial D$ of a bounded convex domain $D \subset \mathbb R^n$. However, I found that such distributions can exist only if $\partial D$ is an ellipsoid. This result gives a new proof of the abovementioned special case of Arnold's conjecture.

Mathematics; Convex and discrete geometry; Functional analysis

video/mp4

Author affiliation: Stockholm University

2020-09-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/76018

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/