On the hyperspace of rotors in convex polygons

L.E.Bazylevych

A rotor in a polygon is a closed convex curve that can be completely rotated inside this polygon so that, in all its positions, it touches all the sides of the polygon. We prove that the hyperspace of all rotors (respectively, of all smooth rotors) in a regular polygon is homeomorphic to the Hilbert cube (respectively, the separable Hilbert space). In the case when the polygon is a square (i.e. for convex curves of constant width) we show that the hyperspace of spherical rotors is homeomorphic to the set of finite sequences in the Hilbert cube.

1. Nadler S.B., Quinn Jr., J., Stavrokas N.M. Hyperspace of compact convex sets, Pacif. J. Math. 83(1979), 441--462.

2. Montejano L. The hyperspace of compact convex subsets of an open subset of $ R^n$, Bull. Pol. Acad. Sci. Math. V.35, no 11-12(1987), 793--799.

3. L.E. Bazylevych, On the hyperspace of strictly convex bodies, Mat. studii. 2(1993), 83--86.

4. Zarichnyi M., Ivanov S. Hyperspaces of compact convex subsets in the Tychonov cube, Ukr. mat. zh. 53 (2001), N 5, 698--701; translation in Ukrainian Math. J. 53 (2001), no. 5, 809--813.

5. Goldberg M. Circular-arc rotors in regular polygons. Amer. Math. Monthly 55(1948), 393--402.

6. Goldberg M. Trammel rotors in regular polygons. Amer. Math. Monthly 64(1957), 71--78.

7. Goldberg M. Rotors in spherical polygons, J. Math. Phys. 30(1952), 235--244.

8. Fujiwara M. Uber die einem Vielecke eigenschreibenem und umdrehbaren konvexen geslossenen Kurven, T\^ohoku Science Reports, 4(1915), 43--55.

9. Bazylevych L.E. Topology of the hyperspace of convex bodies of constant width, Mat. zametki. 62 (1997), 813--819 (in Russian).

10. Bazylevych L.E. Zarichnyi M.M. On convex bodies of constant width, Topol. Appl. (to appear).

11. Schneider R. Gleitkorper in konvexen Polytopen, J. Reine Angew. Math. 248 (1971), 193-220.

12. Engelking R. Dimension theory. A revised and enlarged translation of ``Teoria wymiaru'', Warszawa 1977, by the author. (English) North-Holland Mathematical Library. Vol. 19. Amsterdam, Oxford, New York: North-Holland Publishing Company. Warszawa: PWN - Polish Scientific Publishers. X, 314p.

13. Bessaga C., Pe czy\'nski A. Selected topics in infinite-dimensional topology.- Monografie Matematyczne, 58, Warsaw: PWN, 1975.

14. Meissner E. Drei Gipsmodelle von Fl\"achen kontante Breite, Z. Math. Phys. 60(1912), 92--94.

15. Blaschke W. Konvexe Bereiche gegebener konstanten Breite und kleinsten Inhalts, Math. Ann. 76 (1915), 504--513.

16. Kupitz Y., Martini H., Wegner B. A linear-time construction of Reuleaux polygons, Beitr\"age zur Algebra und Geometrie, 37 (1996), no 2, 415--427.

17. Dobrowolski T. The compact $Z$-set property in convex sets, Topol. Appl. 23(1986), 163--172.

18. Curtis D., Dobrowolski T., Mogilski J. Some applications of the topological characterizations of the sigma-compact spaces $l_{f}^2$ and $\Sigma$, Trans. Am. Math. Soc. 284(1984), 837--846.

19. Chapman T. Lectures on Hilbert cube manifolds, CBMS Regional Conference Series in Math. 28(1976).

20. Ara\'ujo P.V. Representation of curves of constant width in the hyperbolic plane. Port. Math. 55, no~3(1998), 355--372.

	On the hyperspace of rotors in convex polygons
Author	L.E.Bazylevych mzar@litech.lviv.ua National University ``Lviv Polytechnica''
Abstract	A rotor in a polygon is a closed convex curve that can be completely rotated inside this polygon so that, in all its positions, it touches all the sides of the polygon. We prove that the hyperspace of all rotors (respectively, of all smooth rotors) in a regular polygon is homeomorphic to the Hilbert cube (respectively, the separable Hilbert space). In the case when the polygon is a square (i.e. for convex curves of constant width) we show that the hyperspace of spherical rotors is homeomorphic to the set of finite sequences in the Hilbert cube.
Keywords	hyperspace, rotor, convex polygon
DOI	doi:10.30970/ms.26.1.49-54
Reference	1. Nadler S.B., Quinn Jr., J., Stavrokas N.M. Hyperspace of compact convex sets, Pacif. J. Math. 83(1979), 441--462. 2. Montejano L. The hyperspace of compact convex subsets of an open subset of $ R^n$, Bull. Pol. Acad. Sci. Math. V.35, no 11-12(1987), 793--799. 3. L.E. Bazylevych, On the hyperspace of strictly convex bodies, Mat. studii. 2(1993), 83--86. 4. Zarichnyi M., Ivanov S. Hyperspaces of compact convex subsets in the Tychonov cube, Ukr. mat. zh. 53 (2001), N 5, 698--701; translation in Ukrainian Math. J. 53 (2001), no. 5, 809--813. 5. Goldberg M. Circular-arc rotors in regular polygons. Amer. Math. Monthly 55(1948), 393--402. 6. Goldberg M. Trammel rotors in regular polygons. Amer. Math. Monthly 64(1957), 71--78. 7. Goldberg M. Rotors in spherical polygons, J. Math. Phys. 30(1952), 235--244. 8. Fujiwara M. Uber die einem Vielecke eigenschreibenem und umdrehbaren konvexen geslossenen Kurven, T\^ohoku Science Reports, 4(1915), 43--55. 9. Bazylevych L.E. Topology of the hyperspace of convex bodies of constant width, Mat. zametki. 62 (1997), 813--819 (in Russian). 10. Bazylevych L.E. Zarichnyi M.M. On convex bodies of constant width, Topol. Appl. (to appear). 11. Schneider R. Gleitkorper in konvexen Polytopen, J. Reine Angew. Math. 248 (1971), 193-220. 12. Engelking R. Dimension theory. A revised and enlarged translation of ``Teoria wymiaru'', Warszawa 1977, by the author. (English) North-Holland Mathematical Library. Vol. 19. Amsterdam, Oxford, New York: North-Holland Publishing Company. Warszawa: PWN - Polish Scientific Publishers. X, 314p. 13. Bessaga C., Pe czy\'nski A. Selected topics in infinite-dimensional topology.- Monografie Matematyczne, 58, Warsaw: PWN, 1975. 14. Meissner E. Drei Gipsmodelle von Fl\"achen kontante Breite, Z. Math. Phys. 60(1912), 92--94. 15. Blaschke W. Konvexe Bereiche gegebener konstanten Breite und kleinsten Inhalts, Math. Ann. 76 (1915), 504--513. 16. Kupitz Y., Martini H., Wegner B. A linear-time construction of Reuleaux polygons, Beitr\"age zur Algebra und Geometrie, 37 (1996), no 2, 415--427. 17. Dobrowolski T. The compact $Z$-set property in convex sets, Topol. Appl. 23(1986), 163--172. 18. Curtis D., Dobrowolski T., Mogilski J. Some applications of the topological characterizations of the sigma-compact spaces $l_{f}^2$ and $\Sigma$, Trans. Am. Math. Soc. 284(1984), 837--846. 19. Chapman T. Lectures on Hilbert cube manifolds, CBMS Regional Conference Series in Math. 28(1976). 20. Ara\'ujo P.V. Representation of curves of constant width in the hyperbolic plane. Port. Math. 55, no~3(1998), 355--372.
Pages	49-54
Volume	26
Issue	1
Year	2006
Journal	Matematychni Studii
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