NDLI: Good and Bad Radii of Convex Polygons

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Gritzmann, Peter Habsieger, Laurent Klee, Victor
Copyright Year	1991
Abstract	The radii considered here are the inradius $\rho $, the circumradius R, the diameter $\delta $, and the width $\Delta $. The convex polygons in question have their vertices at points of the integer lattice in $\mathbb{R}^2 $, and their radii are measured with respect to an $\ell ^p $ norm. Computation of these radii for convex polygons (and of their higher-dimensional analogues for convex polytopes) is of interest in connection with a number of applications, and may be regarded as a basic problem in computational geometry. The terms good radius and bad radius refer to the existence or nonexistence of a rationalizing polynomiala nonconstant rational polynomial q such that $q(\varphi (C))$ is rational whenever C is a convex lattice polygon and $\varphi $ is the radius function in question. When a radius is good, the polynomial is a tool for implicit computation of the radius in the binary model of computation; otherwise it seems to be necessary to resort to approximation. It is proved here that all four radii are good when $p \in \{ 1,\infty \} $, while $\delta $ is good when p is an integer and $\Delta $ is good when $p / (p - 1)$ is an integer. Thus $\delta $ and $\Delta $ are both good when $p = 2$, and it turns out that R is also good in this case. However, the main results are that r is bad when $p = 2$ and R is bad for each integer $p \geq 3$.
Starting Page	395
Ending Page	403
Page Count	9
File Format	PDF
ISSN	00975397
DOI	10.1137/0220024
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2
Volume Number	20
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-31
Access Restriction	Subscribed
Subject Keyword	inradius implicit computation Linear programming circumradius Irrationality; linear independence over a field Inequalities and extremum problems convex lattice polygon algebraic number rationalizing polynomial diameter Polynomials: factorization Computer graphics; computational geometry width Convex sets in $2$ dimensions polarity
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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