Skip to main content
SpringerLink
Log in

Über die Kantenlängensumme von Dreieckspolyedern

On the total length of the edges of a polyhedron with triangular faces

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In 1953L. Fejes Tóth published the following conjecture: Among all triangle-faced convex polyhedra with a given inradius the regular tetrahedron and octahedron have the least possible total edge-length. In the following this conjecture is proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literatur

  1. Aberth, O.: An isoperimetric inequality for polyhedra and its application to an extremum problem. Proc. London Math. Soc. (3)13, 322–336 (1963).

    Google Scholar 

  2. Aberth, O.: An inequality for convex polyhedra. J. London Math. Soc. (2)6, 382–384 (1973).

    Google Scholar 

  3. Besicovitch, A. S., andH. G. Eggleston: The total length of the edges of a polyhedron. Quart. J. Math. Oxford (2)8, 172–190 (1957).

    Google Scholar 

  4. Coxeter, H. S. M., andL. Fejes Tóth: The total length of the edges of a non-Euclidean polyhedron with triangular faces. Quart. J. Math. Oxford (2)14, 273–284 (1963).

    Google Scholar 

  5. Fejes Tóth, L.: On the total length of the edges of a polyhedron. Norske Vid. Selsk. Forhdl. Trondheim21, 32–34 (1948).

    Google Scholar 

  6. Fejes Tóth, L.: Lagerungen in der Ebene, auf der Kugel und im Raum. Berlin-Heidelberg-New York: Springer. 1953, (2. Aufl. 1972).

    Google Scholar 

  7. Florian, A.: Integrale auf konvexen Polyedern. Period. Math. Hung.1, 243–278 (1971).

    Google Scholar 

  8. Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Berlin-Göttingen-Heidelberg: Springer. 1957.

    Google Scholar 

  9. Kömhoff, M.: An isoperimetric inequality for convex polyhedra with triangular faces. Canad. Math. Bull.11, 723–727 (1968).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. II. Lehrkanzel für Mathematik, Universität Salzburg, Petersbrunnstraße 19, A-5020, Salzburg, Österreich

    Johann Linhart

Authors
  1. Johann Linhart
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Linhart, J. Über die Kantenlängensumme von Dreieckspolyedern. Monatshefte für Mathematik 83, 25–36 (1977). https://doi.org/10.1007/BF01303009

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01303009

Search

Navigation