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Continuity properties of k-plane integrals and Besicovitch sets

  • K. J. Falconer (a1)

Let Π be a k-dimensional subspace of Rn(n ≥ 2) and let Π denote its orthogonal complement. If xRn we shall write x = x0 + x with x0 ∈ Πand x ∈ Π . If f(x) is a real measurable function on Rn, the k-plane integral F(Π,x )is defined as the integral of f over the affine subspace Π + x with respect to k-dimensional Lebesgue measure (assuming that the integral exists). If k = 1 we get the x-ray transform that arises in the problem of radiographic reconstruction, and if k = n − 1, the k-plane integral is the usual projection or Radon transform. The paper by Smith, Solmon and Wagner (4) contains a survey of results on k-plane integrals. Here we shall be interested in the behaviour of the F (Π, x ) regarded as a function of x for fixed Π for various classes of function f. We shall obtain some surprisingly strong results on the continuity and differentiability of F(Π,x) with respect to x for almost all Π (in the sense of the appropriate Haar measure). As will be seen the dimensions n and k have a crucial effect on what may be said, and most of our results will be confined to the cases where k > ½n.

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(1) A. S. Besicovitch On Kakeya's problem and a similar one. Math. Z. 27 (1928), 312320.

(3) B. Fisher On a problem of Besicovitch. Amer. Math. Monthly 80 (1975), 785787.

(4) K. T. Smith , D. C. Solmon and S. L. Wagner Practical and mathematical aspects of the problem of reconstructing objects from radiographs. Bull. Amer. Math. Soc. 83 (1977), 12271270.

(5) D. C. Solmon The X-ray transform. J. Math. Anal. Appl. 56 (1976), 6183.

(6) D. C. Solmon A note on k-plane integral transforms. J. Math. Anal. Appl. 71 (1979), 351358.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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