Area-perimeter duality in polygon spaces


  • Giorgi Khimshiashvili
  • Gaiane Panina
  • Dirk Siersma



Two natural foliations, guided by area and perimeter, of the configurations spaces of planar polygons are considered and the topology of their leaves is investigated in some detail. In particular, the homology groups and the homotopy type of leaves are determined. The homology groups of the spaces of polygons with fixed area and perimeter are also determined. Besides, we extend the classical isoperimetric duality to all critical points. In conclusion a few general remarks on dual extremal problems in polygon spaces and beyond are given.


El Kacimi Alaoui, A., and Zeggar, A., Area and perimeter foliation on spaces of polygons, Graduate J. Math. 4 (2019), no. 1, 18–29.

Giorgadze, G., and Khimshiashvili, G., On area foliation of triangles, Bull. Georgian Natl. Acad. Sci. (N.S.) 14 (2020), no. 1, 15–22.

Gordon, J., Panina, G., and Teplitskaya, Y., Polygons with prescribed edge slopes: configuration space and extremal points of perimeter, Beitr. Algebra Geom. 60 (2019), no. 1, 1–15.

Khimshiashvili, G., Panina, G., and Siersma, D., Equilibria of three constrained point charges, J. Geom. Phys. 106 (2016), 42–50.

Khimshiashvili, G., Panina, G., and Siersma, D., Extremal areas of polygons with fixed perimeter, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 481 (2019), 136–145.

Leichtweiss, K., Konvexe Mengen, Hochschultext. Springer-Verlag, Berlin-New York, 1980.

Mamaev, D., Oriented Area as a Morse Function on Configuration Spaces of Necklaces, arXiv:2001.02707.

Panina, G., and Khimshiashvili, G., Cyclic polygons are critical points of area. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 360 (2008), 238–245.

Panina, G., and Zhukova, A., Morse index of a cyclic polygon, Cent. Eur. J. Math. 9 (2011), no. 2, 364–377.

Siersma, D., The monodromy of a series of hypersurface singularities, Comment. Math. Helv. 65 (1990), no. 2, 181–197.

Tibăr, M., Bouquet decomposition of the Milnor fibre. Topology 35 (1996), no. 1, 227–241.

Zhukova, A., Morse index of a cyclic polygon II, St. Petersburg Math. J. 24 (2013), no. 3, 461–474.



How to Cite

Khimshiashvili, G. ., Panina, G. ., & Siersma, D. . (2021). Area-perimeter duality in polygon spaces. MATHEMATICA SCANDINAVICA, 127(2), 252–263.