Domains with algebraic X-ray transform

Mark Agranovsky

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Koldobsky, Merkurjev and Yaskin proved in (Koldobsky in Adv Math 320:876-886, 2017) that given a convex body K⊂Rn,n is odd, with smooth boundary, such that the volume of the intersection K∩ L of K with a hyperplane L⊂ Rn (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.

Original languageEnglish
Article number60
JournalAnalysis and Mathematical Physics
Volume12
Issue number2
DOIs
StatePublished - Apr 2022

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Keywords

  • Algebraic hypersurface
  • Convex domains
  • Ellipsoid
  • Volumes
  • X-ray transform

Fingerprint

Dive into the research topics of 'Domains with algebraic X-ray transform'. Together they form a unique fingerprint.

Cite this