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See e. g.D'Arcy W. Thompson,On growth and form. II (Second edition, Cambridge, 1952), where we read on p. 527: “The investigation of he ends of the cell was a more difficult matter than that of its sides, and came later.” However, the “problem of the ends” may be considered today as a simple school-problem, while the “problem of the sides” is a question not yet setlled in its full generality. We may suppose that the authours mentioned above restricted their attention to congruent cells. — As to the problem of the bee's cell it is a strong restriction to consider first the problem of the sides and then the problem of the ends only for hexagonal prisms (under further restrictions). It would be desirable to consider the whole problem under far moe general conditions.
Special cases of this theorem can be found in my bookLagerungen in der Ebene, auf der Kugel und im Raum (Berlin-Göttingen-Heidelberg, 1953, III, § 9, and V. § 9) but without proof of the convexity of the functionU forK>0.
Note that the vertices of the cells lying on the boundary of a particular cell are considered as to the vertices of this cell too. Thus a cell of a tessellation may have more vertices (sides) than it has effectively as a single polygon.
The casen=3 has been settled byHeppes.
Cf. e. g. pp. 551–552 of the book quoted in footnote
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Tóth, L.F. On shortest nets with meshes of equal area. Acta Mathematica Academiae Scientiarum Hungaricae 11, 363–370 (1960). https://doi.org/10.1007/BF02020950
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DOI: https://doi.org/10.1007/BF02020950