Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T21:16:23.619Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  26 February 2010

G. Ewald ,
D. G. Larman  and
C. A. Rogers
G. Ewald
Affiliation:
The University of the Ruhr, Bochum, andUniversity College, London.
D. G. Larman
Affiliation:
The University of the Ruhr, Bochum, andUniversity College, London.
C. A. Rogers
Affiliation:
The University of the Ruhr, Bochum, andUniversity College, London.
Get access

Extract

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.

Type
Research Article
Information
Mathematika , Volume 17 , Issue 1 , June 1970 , pp. 1 - 20
Copyright
Copyright © University College London 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.McMinn, T. J., “On the line segments of a convex surface in E 3”, Pacific J. Math., 10 (1960), 943–946.CrossRefGoogle Scholar
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), 24–25.CrossRefGoogle Scholar
3.Ewald, G., “Über die Schattengrenzen konvexer Körper”, Abh. Math. Sem. Univ. Hamburg, 27 (1964), 167–170.CrossRefGoogle Scholar
4.Klee, V. L., “Can the boundary of a d-dimensional convex body contain segments in all directions?”, Amer. Math. Monthly, 76 (1969), 408–410.CrossRefGoogle Scholar
5.Hodge, W. V. D.Pedoe, D., Algebraic Geometry, I (Cambridge, 1947), Ch. 7.Google Scholar
6.Macbeath, A. M., “A theorem on non-homogeneous lattices”, Annals of Math., (2), 56 (1952), 269–293.CrossRefGoogle Scholar
7.Bonnesen, T.Fenchel, W., Konvexe Körper Ergebuisse d. Math., (Berlin, 1934; New York, 1948), 52.Google Scholar
8.Rogers, C. A.Shephard, G. C., “The difference body of a convex body”, Arch Math., 8 (1957), 220–233.CrossRefGoogle Scholar
9.King, R. H., “Tame Cantor sets in E 3”, Pacific J. Math., 11 (1961), 435–446.Google Scholar