# The metric average of 1D Compact sets

R. Baier, N. Dyn, E. Farkhi:
**The metric average of 1D Compact sets**

in:
*Approximation Theory X: Volume 1. Abstract and Classical Analysis,*
*
C. K. Chui, L. L. Schumaker, J. Stöckler
(eds.)
*

*Innovations in Applied Mathematics*, Vanderbuilt University Press, Nashville, TN, 2002, 9 - 22

MR Nummer:
1924849

Zentralblattnummer:
1043.65034

Keywords: metric average; cancellation property; finite union of intervals; compact sets; algorithm

Mathematics Subject Classification Code: 52A27 (65D18 65G30)

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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.

## Contents:

1. | Introduction |

2. | Definitions and Notation |

3. | Properties of the Metric Average |

4. | Algorithm for Computing the Metric Average |

5. | Cancellation Property |

6. | Proofs |