## Symmetric products of the line: Embeddings and retractions

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- by Leonid V. Kovalev PDF
- Proc. Amer. Math. Soc.
**143**(2015), 801-809 Request permission

## Abstract:

The $n$th symmetric product of a metric space is the set of its nonempty subsets with cardinality at most $n$, equipped with the Hausdorff metric. We prove that every symmetric product of the line is an absolute Lipschitz retract and admits a bi-Lipschitz embedding into a Euclidean space of sufficiently high dimension.## References

- Frederick Justin Almgren Jr.,
*The homotopy groups of the integral cycle groups*, Topology**1**(1962), 257–299. MR**146835**, DOI 10.1016/0040-9383(62)90016-2 - Robert N. Andersen, M. M. Marjanović, and Richard M. Schori,
*Symmetric products and higher-dimensional dunce hats*, Topology Proc.**18**(1993), 7–17. MR**1305120** - G. Bianchi and M. Longinetti,
*Reconstructing plane sets from projections*, Discrete Comput. Geom.**5**(1990), no. 3, 223–242. MR**1036872**, DOI 10.1007/BF02187787 - Marina Borovikova and Zair Ibragimov,
*The third symmetric product of $\Bbb R$*, Comput. Methods Funct. Theory**9**(2009), no. 1, 255–268. MR**2478275**, DOI 10.1007/BF03321726 - Marina Borovikova, Zair Ibragimov, and Hassan Yousefi,
*Symmetric products of the real line*, J. Anal.**18**(2010), 53–67. MR**2850235** - Karol Borsuk and Stanislaw Ulam,
*On symmetric products of topological spaces*, Bull. Amer. Math. Soc.**37**(1931), no. 12, 875–882. MR**1562283**, DOI 10.1090/S0002-9904-1931-05290-3 - R. Bott,
*On the third symmetric potency of $S_1$*, Fund. Math.**39**(1952), 264–268 (1953). MR**54954**, DOI 10.4064/fm-39-1-264-268 - Alexander Brudnyi and Yuri Brudnyi,
*Methods of geometric analysis in extension and trace problems. Volume 2*, Monographs in Mathematics, vol. 103, Birkhäuser/Springer Basel AG, Basel, 2012. MR**2868143** - Sara Brunetti and Alain Daurat,
*Stability in discrete tomography: some positive results*, Discrete Appl. Math.**147**(2005), no. 2-3, 207–226. MR**2127075**, DOI 10.1016/j.dam.2004.09.012 - Dmitri Burago, Yuri Burago, and Sergei Ivanov,
*A course in metric geometry*, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR**1835418**, DOI 10.1090/gsm/033 - L. Capogna, J. T. Tyson and S. Wenger (eds.),
*AimPL: Mapping theory in metric spaces*, available at http://aimpl.org/mappingmetric - Camillo De Lellis and Emanuele Nunzio Spadaro,
*$Q$-valued functions revisited*, Mem. Amer. Math. Soc.**211**(2011), no. 991, vi+79. MR**2663735**, DOI 10.1090/S0065-9266-10-00607-1 - Herbert Federer and Wendell H. Fleming,
*Normal and integral currents*, Ann. of Math. (2)**72**(1960), 458–520. MR**123260**, DOI 10.2307/1970227 - Richard J. Gardner,
*Geometric tomography*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR**2251886**, DOI 10.1017/CBO9781107341029 - R. J. Gardner and Peter Gritzmann,
*Discrete tomography: determination of finite sets by X-rays*, Trans. Amer. Math. Soc.**349**(1997), no. 6, 2271–2295. MR**1376547**, DOI 10.1090/S0002-9947-97-01741-8 - Jordan Goblet,
*Lipschitz extension of multiple Banach-valued functions in the sense of Almgren*, Houston J. Math.**35**(2009), no. 1, 223–231. MR**2491878** - A. Heppes,
*On the determination of probability distributions of more dimensions by their projections*, Acta Math. Acad. Sci. Hungar.**7**(1956), 403–410 (English, with Russian summary). MR**85646**, DOI 10.1007/BF02020535 - Alejandro Illanes and Sam B. Nadler Jr.,
*Hyperspaces*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 216, Marcel Dekker, Inc., New York, 1999. Fundamentals and recent advances. MR**1670250** - Urs Lang and Thilo Schlichenmaier,
*Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions*, Int. Math. Res. Not.**58**(2005), 3625–3655. MR**2200122**, DOI 10.1155/IMRN.2005.3625 - A. Rényi,
*On projections of probability distributions*, Acta Math. Acad. Sci. Hungar.**3**(1952), 131–142 (English, with Russian summary). MR**53422**, DOI 10.1007/bf02022515 - R. M. Schori,
*Hyperspaces and symmetric products of topological spaces*, Fund. Math.**63**(1968), 77–88. MR**232336**, DOI 10.4064/fm-63-1-77-88

## Additional Information

**Leonid V. Kovalev**- Affiliation: Department of Mathematics, 215 Carnegie, Syracuse University, Syracuse, New York 13244-1150
- MR Author ID: 641917
- Email: lvkovale@syr.edu
- Received by editor(s): December 7, 2012
- Received by editor(s) in revised form: June 5, 2013
- Published electronically: October 15, 2014
- Additional Notes: This research was supported by the NSF grant DMS-0968756.
- Communicated by: Jeremy Tyson
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**143**(2015), 801-809 - MSC (2010): Primary 30L05; Secondary 54E40, 54B20, 54C15, 54C25
- DOI: https://doi.org/10.1090/S0002-9939-2014-12280-5
- MathSciNet review: 3283666