Quantifying curvelike structures of measures by using L2 Jones quantities
Copyright policy )Quantifying curvelike structures of measures by using L2 Jones quantities
AbstractWe study the curvelike structure of special measures on ℝn in a multiscale fashion. More precisely, we consider the existence and construction of a sufficiently short curve with a sufficiently large measure. Our main tool is an L2 variant of Jones' β numbers, which measure the scaled deviations of the given measure from a best approximating line at different scales and locations. The Jones function is formed by adding the squares of the L2 Jones numbers at different scales and the same location. Using a special L2 Jones function, we construct a sufficiently short curve with a sufficiently large measure. The length and measure estimates of the underlying curve are expressed in terms of the size of this Jones function. © 2003 Wiley Periodicals, Inc.
- New York University United States
Length, area, volume, other geometric measure theory, curvelike structures, Hausdorff and packing measures, Information storage and retrieval of data, Capacity and harmonic measure in the complex plane, Geometric measure and integration theory, integral and normal currents in optimization, rectifiable curves, data sets, approximation
Length, area, volume, other geometric measure theory, curvelike structures, Hausdorff and packing measures, Information storage and retrieval of data, Capacity and harmonic measure in the complex plane, Geometric measure and integration theory, integral and normal currents in optimization, rectifiable curves, data sets, approximation
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