Real Analysis Exchange

Pseudo-characteristic functions for convex polyhedra.

W. A. Beyer, Stephen L. Judd, and Johndale C. Solem

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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.

Article information

Real Anal. Exchange, Volume 29, Number 2 (2003), 821-835.

First available in Project Euclid: 7 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 52B11: $n$-dimensional polytopes 51M15: Geometric constructions

Convex Polyhedra Euclidean N-space simplex characteristic functions psuedo-characteristic functions


Beyer, W. A.; Judd, Stephen L.; Solem, Johndale C. Pseudo-characteristic functions for convex polyhedra. Real Anal. Exchange 29 (2003), no. 2, 821--835.

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