Abstract
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.
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