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The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space

  • G. Ewald (a1), D. G. Larman (a1) and C. A. Rogers (a1)

It is well-known and easy to prove that the maximal line segments on the boundary of a convex domain in the plane are countable. T. J. McMinn [1] has shown that the end-points of the unit vectors drawn from the origin in the directions of the line segments lying on the surface of a convex body in 3-dimensional Euclidean space E3 form a set of σ-finite linear Hausdorff measure on the 2-dimensional surface of the unit ball. A. S. Besicovitch [2] has given a simpler proof of McMinn's result. W. D. Pepe, in a paper to appear in the Proc. Amer. Math. Soc., has extended the result to E4. In this paper we generalize McMinn's result to En by use of Besicovitch's method, proving:

THEOREM 1. If K is a convex body in En, the set S, of end-points of the vectors drawn from origin in the directions of the line segments lying on the surface of K, is a set of σ-finite (n − 2)-dimensional Hausdorff measure on the (n − 1)-dimensional surface of the unit ball.

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1.McMinn, T. J., “On the line segments of a convex surface in E 3”, Pacific J. Math., 10 (1960), 943946.
2.Besicovitch, A. S., “Onthe set of directions of linear segments on a convex surface”, Proc. Amer. Math. Soc. Symp. Pure Math. (Convexity), 7 (1963), 2425.
3.Ewald, G., “Über die Schattengrenzen konvexer Körper”, Abh. Math. Sem. Univ. Hamburg, 27 (1964), 167170.
4.Klee, V. L., “Can the boundary of a d-dimensional convex body contain segments in all directions?”, Amer. Math. Monthly, 76 (1969), 408410.
5.Hodge, W. V. D.Pedoe, D., Algebraic Geometry, I (Cambridge, 1947), Ch. 7.
6.Macbeath, A. M., “A theorem on non-homogeneous lattices”, Annals of Math., (2), 56 (1952), 269293.
7.Bonnesen, T.Fenchel, W., Konvexe Körper Ergebuisse d. Math., (Berlin, 1934; New York, 1948), 52.
8.Rogers, C. A.Shephard, G. C., “The difference body of a convex body”, Arch Math., 8 (1957), 220233.
9.King, R. H., “Tame Cantor sets in E 3”, Pacific J. Math., 11 (1961), 435446.
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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