Pseudo-characteristic functions for convex polyhedra.

Abstract

An algorithm is given for constructing polynomials that determine approximately whether a point $p$ is inside or outside a given polyhedron $C_n$ in Euclidean $n$-dimensional space. The polynomials are of degree $2r$, where $r$ is a positive integer and the order of the approximation can be made arbitrarily small by taking $r$ sufficiently large. For $n=2$, the square, triangle, trapezoid, and pentagon are used as examples. For $n=3$ and $n=4$, the tetrahedron and equilateral simplex are used as examples. We conjecture that the center of mass of the region determined by the approximating polynomial is the same for all values of $r$, and hence coincides with the center of the polyhedra.