Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T12:38:27.912Z Has data issue: false hasContentIssue false
  • English
  • Français

Helly's Theorems on Convex Domains and Tchebycheff's Approximation Problem

Published online by Cambridge University Press:  20 November 2018

Hans Rademacher  and
I. J. Schoenberg
Hans Rademacher
Affiliation:
The University of Pennsylvania
I. J. Schoenberg
Affiliation:
The University of Pennsylvania
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Professor Dresden called to our attention the following theorem :

If S1, S2, … , Sm are m line segments parallel to the y-axis, all of equal lengths, whose projections on the x-axis are equally spaced, and if we assume that a straight line can be made to intersect every set of three among these segments, then there exists a straight line intersecting all the segments.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 2 , 1950 , pp. 245 - 256
Copyright
Copyright © Canadian Mathematical Society 1950

References

[1] Alexandroff, P. and Hopf, H., Topologie, vol. 1, Berlin, 1935.Google Scholar
[2] Blumenthal, L. M., Metric methods in linear inequalities, Duke Math. J., vol. 15 (1948), 955–966.Google Scholar
[3] Dines, L. L. and McCoy, N. H., On linear inequalities, Trans. Roy. Soc. Can., Third Series, Sec. III , vol. 27 (1933), 37–70.Google Scholar
[4] Helly, E., Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jahresbericht der deutschen Mathematiker Vereinigung, vol. 32 (1923), 175–176.Google Scholar
[5] Kirchberger, P., Über Tschebyschefsche Annaherungsmethoden, Math. Ann., vol. 57 (1903), 509–540. The same paper appeared also in more elaborate form (96 pages) in 1902 as a doctoral dissertation written under Hilbert's guidance.Google Scholar
[6] Konig, D., Über konvexe Körper, Math. Zeit., vol. 14 (1922), 208–210.Google Scholar
[7] Radon, J., Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten, Math. Ann., vol. 83 (1921), 113–115.Google Scholar
[8] Tarski, A., A decision method for elementary algebra and geometry, The Rand Corporation, 1948, 57 pages.Google Scholar
[9] de la Vallée Poussin, Ch. J., Leçons sur l' approximation des fonctions d'une variable reçlle, Paris, 1919.Google Scholar