The Metric Average of 1D Compact Sets

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Baier, Robert ; Dyn, N. ; Farkhi, E.:
The Metric Average of 1D Compact Sets.
In: Chui, Charles K. ; Schumaker, Larry L. ; Stöckler, Joachim (Hrsg.): Approximation Theory X : selections of papers that were presented at the Tenth International Conference on Approximation Theory, held in St. Louis, Missouri, in March 2001. Volume 1. Abstract and classical analysis. - Nashville : Vanderbilt Univ. Press , 2002 . - S. 9-22 . - (Innovations in Applied Mathematics )
ISBN 0-8265-1415-4

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Abstract

We study properties of a binary operation between two compact sets depending on a weight in [0,1], termed metric average. The metric average is used in spline subdivision schemes for compact sets in |R^n, instead of the Minkowski convex combination of sets, to retain non-convexity, see N. Dyn, E. Farkhi, ``Spline subdivision schemes for compact sets with metric averages", Trends in Approximation Theory (2001).
Some properties of the metric average of sets in |R, like the cancellation property and the linear behavior of the Lebesgue measure of the metric average with respect to the weight, are proven. We present an algorithm for computing the metric average of two compact sets in |R, which are finite unions of intervals, as well as an algorithm for reconstructing one of the metric average's operands, given the second operand, the metric average and the weight.