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The geometry of fractal sets. (English) Zbl 0587.28004
Cambridge Tracts in Mathematics, 85. Cambridge etc.: Cambridge University Press. XIV, 162 p. £17.50; $ 32.50 (1985).
This book provides a rigorous self-contained account of one of the most active branches of geometric measure theory, the geometrical study of sets of integral and fractional Hausdorff dimension. The major object of this investigation is the existence of tangents of such sets; measure dimensional properties of their projections in various directions. In the case of sets of integral dimension the difference between ”regular” (in Besicovitch sense) ”curve-like” sets and irregular ”dust-like” sets are investigated in great detail. One chapter gives a quite complete account on the Kakeya’s problem, i.e. the problem of finding the area of the smallest convex set inside which a unit segment can be reversed. Starting with the classical solution of this problem due to Besicovitch, the chapter ends with more recent results in this direction exhibiting another approach to this set of problems based on a certain geometric duality. The final part of the book includes a great variety of examples to which the general theory is applicable. We mention various examples of curves of fractional dimension, attractors, examples from number theory, convex analysis, Brownian motion etc.
The book is supplemented by a good bibliography which together with classical literature on the subject includes many current publications.
Reviewer: D.Khavinson

MSC:
28A75 Length, area, volume, other geometric measure theory
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
MathOverflow Questions:
The product of two Hausdorff measures