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The volume function defined by a domain in Euclidean space $\mathbb{R}^n$ isthe function on the space of affine hyperplanes equal to volumes cut by thesehyperplanes from the domain. The study of these functions originates from theworks of Archimedes and Newton and is closely related to the theory of lacunasof hyperbolic partial differential equations.The volume functions are regularat the hyperplanes of general position with respect to the boundary of the cutdomain. We study their behavior at the non-regular hyperplanes, which areeither tangent to the boundary of the domain at its finite points or haveasymptotic direction. In both cases the local regularity of the restriction ofthe volume function to a local connected component of the set of regular planesdepends on the triviality of a certain relative homology class, the(generalized) even Petrovsky class. In the first case (of finite tangencies)the study of these classes and enumeration of components of regularity (socalled local lacunas) at simple singularities of wave fronts was essentiallydone by V. A. Vassiliev; it is formulated in terms of deformations of simplereal function singularities. We show that the analogous study for asymptotichyperplanes is related in the same way with the study of boundary functionsingularities. We define and calculate local Petrovsky classes for this caseand find lacunas, i.e. the local components of the complement of correspondingdiscriminant in which all local Petrovsky classes are trivial. If component isnot lacuna we prove that the corresponding volume function cannot be algebraic.Also we calculate the local monodromy groups, describing the ramification ofvolume functions.

Lacunas and ramification of volume functions at simple asymptotic hyperplanes and monodromy of boundary function singularities