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Mat. Zametki, 2019, Volume 106, Issue 6, Pages 848–853 (Mi mz12454)  

New Examples of Locally Algebraically Integrable Bodies

V. A. Vassilievab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b National Research University "Higher School of Economics", Moscow

Abstract: Any compact body with regular boundary in ${\mathbb R}^N$ defines a two-valued function on the space of affine hyperplanes: the volumes of the two parts into which these hyperplanes cut the body. This function is never algebraic if $N$ is even and is very rarely algebraic if $N$ is odd: all known bodies defining algebraic volume functions are ellipsoids (and have been essentially found by Archimedes for $N=3$). We demonstrate a new series of locally algebraically integrable bodies with algebraic boundaries in spaces of arbitrary dimensions, that is, of bodies such that the corresponding volume functions coincide with algebraic ones in some open domains of the space of hyperplanes intersecting the body.

Keywords: integral geometry, lacuna, algebraic function, algebraic integrability.

Funding Agency Grant Number
Russian Science Foundation 16-11-10316
This work was supported by the Russian Science Foundation under grant 16-11-10316.


DOI: https://doi.org/10.4213/mzm12454

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English version:
Mathematical Notes, 2019, 106:6, 894–898

Bibliographic databases:

UDC: 517.444
Received: 19.05.2019
Revised: 22.05.2019

Citation: V. A. Vassiliev, “New Examples of Locally Algebraically Integrable Bodies”, Mat. Zametki, 106:6 (2019), 848–853; Math. Notes, 106:6 (2019), 894–898

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