Functions of constant geodesic X-ray transform

Ilmavirta, Joonas; Paternain, Gabriel P.

doi:10.1088/1361-6420/ab0b6f

by Joonas Ilmavirta, Gabriel P. Paternain

Abstract:

We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions of constant geodesic X-ray transform always exist on manifolds with rotational symmetry.

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Reference:

Functions of constant geodesic X-ray transform (Joonas Ilmavirta, Gabriel P. Paternain), Inverse Problems, volume 35, number 6, 2019. [show abstract] [hide abstract] We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions of constant geodesic X-ray transform always exist on manifolds with rotational symmetry. [arXiv]

Bibtex Entry:

@article{constant-xrt,
	author = {Joonas Ilmavirta and Gabriel P. Paternain},
	title = {{Functions of constant geodesic X-ray transform}},
	journal = {Inverse Problems},
	volume = {35},
	number = {6},
	month = may,
	year = {2019},
	arxiv = {1812.03515},
	url={http://users.jyu.fi/~jojapeil/pub/constant-xrt-v10.pdf},
	doi = {10.1088/1361-6420/ab0b6f},
        gsid = {12361788843628514661},
	abstract = {We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions of constant geodesic X-ray transform always exist on manifolds with rotational symmetry.}
}