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Packing smooth curves in Rq

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

J. M. Marstrand
J. M. Marstrand
Affiliation:
School of Mathematics, University Walk, Bristol, BS8 1TW.
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Extract

In [1], [2] Besicovitch showed that it is possible to translate each straight line in the plane so that the union of all the translates has zero plane measure. More recently Besicovitch and Rado [3] and independently Kinney [12] showed that the same can be done with arcs of circles instead of straight lines (see also Davies [6]). Allowing rotations as well as translations, Ward [18] showed that all plane polygonal curves can be “packed” thus (allowing overlapping) into zero plane measure, and then Davies [7], making use of Besicovitch's construction, showed translations alone to be sufficient, although these papers in fact contained stronger results concerning Hausdorff measure; the results were further generalized in [16]. The question has naturally been asked whether the class of all plane rectifiable curves can be packed by isometries (translations and rotations) into zero plane measure, but a special case of the main theorem of the present paper shows that this is impossible. The corresponding question remains open for the much smaller class of algebraic curves, or even conies.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory

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Type
Research Article
Information
Mathematika , Volume 26 , Issue 1 , June 1979 , pp. 1 - 12
Copyright
Copyright © University College London 1979

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