New Examples of Locally Algebraically Integrable Bodies


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Abstract

Any compact body with regular boundary in ℝN 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.

About the authors

V. A. Vassiliev

Steklov Mathematical Institute of Russian Academy of Sciences; National Research University Higher School of Economics

Author for correspondence.
Email: vva@mi-ras.ru
Russian Federation, Moscow, 119991; Moscow, 101000

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