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Domains with algebraic X-ray transform

Domains with algebraic X-ray transform Koldobsky, Merkurjev and Yaskin proved in (Koldobsky in Adv Math 320:876-886, 2017) that given a convex body K⊂Rn,n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K \subset {\mathbb {R}}^n, \ n$$\end{document} is odd, with smooth boundary, such that the volume of the intersection K∩L\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K \cap L$$\end{document} of K with a hyperplane L⊂Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L \subset {\mathbb {R}}^n$$\end{document} (the sectional volume function) depends polynomially on the distance t of L to the origin, then the boundary of K is an ellipsoid. In even dimension, the sectional volume functions are never polynomials in t, nevertheless in the case of ellipsoids their squares are. We conjecture that the latter property fully characterizes ellipsoids and, disregarding the parity of the dimension, ellipsoids are the only convex bodies with smooth boundaries whose sectional volume functions are roots (of some power) of polynomials. In this article, we confirm this conjecture for planar domains, bounded by algebraic curves. A multidimensional version in terms of chords lengths, i.e., of X-ray transform of the characteristic function, is given. The result is motivated by Arnold’s conjecture on characterization of algebraically integrable bodies. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Domains with algebraic X-ray transform

Agranovsky, Mark
Analysis and Mathematical Physics , Volume 12 (2) – Apr 1, 2022
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References (12)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-022-00657-x
Publisher site
See Article on Publisher Site

Abstract

Koldobsky, Merkurjev and Yaskin proved in (Koldobsky in Adv Math 320:876-886, 2017) that given a convex body K⊂Rn,n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K \subset {\mathbb {R}}^n, \ n$$\end{document} is odd, with smooth boundary, such that the volume of the intersection K∩L\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K \cap L$$\end{document} of K with a hyperplane L⊂Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L \subset {\mathbb {R}}^n$$\end{document} (the sectional volume function) depends polynomially on the distance t of L to the origin, then the boundary of K is an ellipsoid. In even dimension, the sectional volume functions are never polynomials in t, nevertheless in the case of ellipsoids their squares are. We conjecture that the latter property fully characterizes ellipsoids and, disregarding the parity of the dimension, ellipsoids are the only convex bodies with smooth boundaries whose sectional volume functions are roots (of some power) of polynomials. In this article, we confirm this conjecture for planar domains, bounded by algebraic curves. A multidimensional version in terms of chords lengths, i.e., of X-ray transform of the characteristic function, is given. The result is motivated by Arnold’s conjecture on characterization of algebraically integrable bodies.

Journal

Analysis and Mathematical Physics – Springer Journals

Published: Apr 1, 2022

Keywords: Convex domains; X-ray transform; Volumes; Algebraic hypersurface; Ellipsoid; 44A12; 51M25

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