# Lacunas and local algebraicity of volume functions

## TL;DR

This paper investigates whether volume functions of bodies in Euclidean space can be locally algebraic near certain hyperplanes, revealing obstructions and providing examples of bodies with locally algebraic volume functions.

## Contribution

It introduces homological and geometric obstructions to local algebraicity of volume functions and constructs examples of bodies with locally algebraic volume functions.

## Key findings

- Homological obstructions prevent local algebraicity in certain cases.
- Geometric restrictions limit when volume functions can be locally algebraic.
- Examples of bodies with locally algebraic volume functions are identified.

## 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