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Isoperimetric problems for tilings

  • L. Fejes Tóth (a1)

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We consider a tiling of a square with convex tiles of areas a1,… aN and perimeters p1,…,pN and pose the following problems.

PROBLEM 1. For given values of a1, …, aN find the minimum of P = p1 + … + pN.

PROBLEM 2. For given values of p1,…,pN find the maximum of A = a1+…+aN.

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References

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1.Bambah, R. P. and Rogers, C. A.. Covering the plane with convex sets. J. London Math. Soc., 27 (1952), 304314.
2.Fejes Tóth, G.. An isoperimetric problem for tessellations. Studia Sci. Math. Hungar., 10 (1975), 171173.
3.Fejes Tóth, L.. Regular Figures (Pergamon Press, 1964).
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