Lacunas and local algebraicity of volume functions

Journal of Singularities
volume 18 (2018), 350-357

Received: 31 January 2018. Accepted: 20 February 2018.

DOI: 10.5427/jsing.2018.18r

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Abstract:

The volume cut off by a hyperplane from a bounded body with smooth boundary in R^{2k} never is an algebraic function on the space of hyperplanes: for k=1 it is the famous lemma XXVIII from Newton's Principia. Following an analogy of these volume functions with the solutions of hyperbolic PDE's, we study the local version of the same problem: can such a volume function coincide with an algebraic one at least in some domains of the space of hyperplanes, intersecting the body? We prove some homological and geometric obstructions to this integrability property. Based on these restrictions, we find a family of examples of such "locally integrable" bodies in Euclidean spaces.

Keywords:

Integral geometry, Picard-Lefschetz theory, lacuna, algebraic function, monodromy, Newton's lemma XXVIII

Mathematical Subject Classification:

14D05, 44A99; 20F55

Author(s) information:

V. A. Vassiliev
Steklov Mathematical Institute and
National Research University Higher School of Economics
Moscow, Russia
email: vva@mi.ras.ru