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On the Total Area of the Faces Of a Four-Dimensional Polytope

Published online by Cambridge University Press:  20 November 2018

L. Fejes Tóth
L. Fejes Tóth*
Affiliation:
University of Veszprérn, Hungary
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Let L be the total length of the edges of a convex polyhedron containing a unit sphere. If the number of edges is small, the edges must be, on the average, comparatively long. If, on the other hand, the edges are short, their number must be great. So the problem arises to find a polyhedron with a possibly small value of L.

By a simple argument the author (5; 6) proved that L > 20 and announced the conjecture that L ≥ 24 with equality only for the cube (of in-radius 1). The same argument shows that for trigonal-faced polyhedra L > 28. This supports the conjecture that for such polyhedra L ≥ 12√6 = 29.4 . . . , with equality only for the tetrahedron and octahedron.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 93 - 99
Copyright
Copyright © Canadian Mathematical Society 1965

References

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