| Title:
|
Symmetric products of the Euclidean spaces and the spheres (English) |
| Author:
|
Chinen, Naotsugu |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
56 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
209-221 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
By $F_n(X)$, $n \geq 1$, we denote the $n$-th symmetric product of a metric space $(X,d)$ as the space of the non-empty finite subsets of $X$ with at most $n$ elements endowed with the Hausdorff metric $d_H$. In this paper we shall describe that every isometry from the $n$-th symmetric product $F_n(X)$ into itself is induced by some isometry from $X$ into itself, where $X$ is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the $n$-th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and present that the $2$nd symmetric product of the plane is bi-Lipschitz equivalent to the 4-dimensional Euclidean space. (English) |
| Keyword:
|
isometry |
| Keyword:
|
symmetric product |
| Keyword:
|
bi-Lipschitz maps |
| Keyword:
|
Euclidean space |
| Keyword:
|
sphere |
| MSC:
|
30C65 |
| MSC:
|
30L10 |
| MSC:
|
54B10 |
| MSC:
|
54B20 |
| MSC:
|
54E35 |
| idZBL:
|
Zbl 06433818 |
| idMR:
|
MR3338733 Reviewed Chinen, Naot |
| DOI:
|
10.14712/1213-7243.2015.118 |
| . |
| Date available:
|
2015-04-25T17:04:38Z |
| Last updated:
|
2017-08-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144241 |
| . |
| Reference:
|
[1] Bandt C.: On the metric structure of hyperspaces with Hausdorff metric.Math. Nachr. 129 (1986), 175–183. Zbl 0609.54008, MR 0864632, 10.1002/mana.19861290116 |
| Reference:
|
[2] Belobrov P.K.: The Čebyšev point of a system of sets.Izv. Vysš. Učebn. Zaved. Matematika 55 (1966), 18–24. Zbl 0192.22501, MR 0208332 |
| Reference:
|
[3] Bestvina M.: $\mathbb R$-trees in Topology, Geometry, and Group Theory.Handbook of Geometric Topology, 55–91, North-Holland, Amsterdam, 2002. MR 1886668 |
| Reference:
|
[4] Borsuk K., Ulam S.: On symmetric products of topological spaces.Bull. Amer. Math. Soc. 37 (1931), 875–882. Zbl 0003.22402, MR 1562283, 10.1090/S0002-9904-1931-05290-3 |
| Reference:
|
[5] Borsuk K.: On the third symmetric potency of the circumference.Fund. Math. 36 (1949), 236–244. Zbl 0039.19301, MR 0035987 |
| Reference:
|
[6] Borovikova M., Ibragimov Z.: The third symmetric product of $ \mathbb R$.Comput. Methods Funct. Theory 9 (2009), 255–268. MR 2478275, 10.1007/BF03321726 |
| Reference:
|
[7] Borovikova M., Ibragimov Z., Yousefi H.: Symmetric products of the real line.J. Anal. 18 (2010), 53–67. Zbl 1239.30030, MR 2850235 |
| Reference:
|
[8] Bott R.: On the third symmetric potency of $S_1$.Fund. Math. 39 (1952), 264–268. Zbl 0050.17801, MR 0054954 |
| Reference:
|
[9] Bridson M.R., Haefliger A.: Metric spaces of non-positive curvature.Grundlehren der Mathematischen Wissenschaften, 319, Springer, Berlin, 1999. Zbl 0988.53001, MR 1744486 |
| Reference:
|
[10] Chinen N., Koyama A.: On the symmetric hyperspace of the circle.Topology Appl. 157 (2010), 2613–2621. Zbl 1205.54018, MR 2725354, 10.1016/j.topol.2010.07.012 |
| Reference:
|
[11] Foertsch T.: Isometries of spaces of convex compact subsets of globally non-positively Busemann curved spaces.Colloq. Math. 103 (2005), 71–84. Zbl 1077.53063, MR 2148951, 10.4064/cm103-1-9 |
| Reference:
|
[12] Illanes A.: Nadler S.B., Jr..{\it Hyperspaces}, Marcel Dekker, New York, 1999. MR 1670250 |
| Reference:
|
[13] Ivanshin P.N., Sosov E.N.: Local Lipschitz property for the Chebyshev center mapping over N-nets.Mat. Vesnik 60 (2008), 9–22. Zbl 1199.54169, MR 2403268 |
| Reference:
|
[14] Kovalev L.V.: Symmetric products of the line: embeddings and retractions.Proc. Amer. Math. Soc. 143 (2015), 801-809. MR 3283666, 10.1090/S0002-9939-2014-12280-5 |
| Reference:
|
[15] Molski R.: On symmetric product.Fund. Math. 44 (1957), 165–170. MR 0092953 |
| Reference:
|
[16] Morton H.R.: Symmetric product of the circle.Proc. Cambridge Philos. Soc. 63 (1967), 349–352. MR 0210096 |
| Reference:
|
[17] Valentine F.A.: Convex Sets.McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. Zbl 0333.52001, MR 0170264 |
| Reference:
|
[18] Wu W.: Note sur les produits essentiels symétriques des espaces topologiques.C.R. Acad. Sci. Paris 224 (1947), 1139–1141. MR 0019914 |
| . |
