Abstract
Out-of-roundness measures in the plane are described geometrically and formulated analytically in terms of optimization theory. The analytical framework that we develop permits a more complete and thorough examination of out-of-roundness measures, and allows these measures to be generalized and extended to general convex bodies in R″. A new classification scheme based on “centers” is used to categorize out-of-roundness measures, which has also proved useful in other contexts. Mathematical characterizations of out-of-roundness measures are then related to the algorithms of mathematical programming which are used to actually compute these points. In addition to this discussion, relevant complexity issues on determining these measures are also introduced. Several alternative out-of-roundness measures are formulated and the geometry of their formulation discussed. This includes symmetry and various gravity measures, related least squares measures, and other more esoteric formulations.
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