Real Analysis Exchange

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.

Article information

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

Dates
First available in Project Euclid: 7 June 2006

https://projecteuclid.org/euclid.rae/1149698545

Mathematical Reviews number (MathSciNet)
MR2083817

Zentralblatt MATH identifier
1072.52002

Citation

Beyer, W. A.; Judd, Stephen L.; Solem, Johndale C. Pseudo-characteristic functions for convex polyhedra. Real Anal. Exchange 29 (2003), no. 2, 821--835. https://projecteuclid.org/euclid.rae/1149698545

References

• Tom M. Apostol, Linear Algebra, John Wiley & Sons, 1997, 150–151.
• Jeff Cheeger and Detlef Gromoll, On the Lower Bound for the Injectivity Radius of 1/4-Pinched Riemannian Manifolds, J. Differential Geometry, 15, no.3 (1979), 437–442.
• Encyclopedic Dictionary of Mathematics, Mathematical Society of Japan, 1977, Convex sets, 304–305, Regular polyhedra, 1103–5110, Massachusetts Institute of Technology.
• M. Jacob and S. Andersson, The Nature of Mathematics and the Mathematics of Nature, Elsevier, 1998
• Joseph O'Rourke, Computational Geometry in C, Cambridge University Press, 1999, 245.
• Elena Prestini, A note on $L_p$ multipliers given by the characteristic function of unbounded polygonal regions of the plane, Boll. Un. Mat. Ital. A., 6 (1984), 125-130.
• Jerome Spanier and Keith B. Oldham, An Atlas of Functions, Hemisphere Publishing Corp., 1989.
• Eric J. Stollnitz, Tony D. Derosse, and David H. Salesin, Wavelets for Computer Graphics, Theory and Applications, Morgan Kaufmann Publishers, Inc., 1996.
• I. M. Yaglom and V. G. Boltyanskiĭ, Convex Figures, Holt, Rinehart and Winston, 1961.