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Literatur
Theorie der vielfachen Kontinuität,Denkschriften der schweizerischen naturforschenden Gesellschaft,38 (1901), pp. 1–237; —Ges. Abh. I (Basel, 1950), pp. 167–387.
See e. g. the remarks on the end of the paper.
L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin-Göttingen-Heidelberg, 1953).
H. S. M. Coxeter,Regular polytopes (London, 1948).
L. Fejes Tóth, On close-packings of spheres in spaces of constant curvature,Publicationes Math. (Debrecen),3 (1953), pp. 158–167.
Compare our proof withM. Goldberg, The isoperimetric problem for polyhedra,Tôhoku Math. J.,42 (1935), pp. 226–236, andH. Hadwiger, Zur isoperimetrischen Ungleichung fürk-dimensionale konvexe Polyeder,Nagoya Math. J.,5 (1953), pp. 39–44.
Such a characterisation is impossible by comparing polyhedra of given number of faces or vertices.
We denote a body and its volume by the same symbol.
The symmetrisation of a tetrahedron and octahedron was already used bySteiner in order to establish the isoperimetric property of {3, 3} and {3, 4}.
A very simple direct proof of this was given byI. Ádám. (See the book quoted in footnote3,L. Fejes Tóth,Lagerungen in der Ebene, auf der Kugel und im Raum (Berlin-Göttingen-Heidelberg, 1953) p. 28.)
Cf. the book cited in3, p. 131.
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Tóth, L.F. Extremum properties of the regular polytopes. Acta Mathematica Academiae Scientiarum Hungaricae 6, 143–146 (1955). https://doi.org/10.1007/BF02021272
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DOI: https://doi.org/10.1007/BF02021272