Computational Methods and Function Theory 9 (2009), No. 1, 255--268
Copyright Heldermann Verlag 2009
Marina Borovikova
mborovikova@fullerton.edu , California State University, Department of Mathematics, 154 McCarthy Hall, Fullerton, CA 92834, U.S.A.
Zair Ibragimov
zibragimov@fullerton.edu , California State University, Department of Mathematics, 154 McCarthy Hall, Fullerton, CA 92834, U.S.A.
We study geometry of the space of all non-empty subsets of the real line R of cardinality at most three endowed with the Hausdorff metric. The space is known to be homeomorphic to the 3-dimensional Euclidean space R3. We prove that it is, in fact, (3+4π)-bi-Lipschitz equivalent to R3. We also show that all isometries of R3 are induced by isometries of R.
Keywords: Symmetric products, isometry, bi-Lipschitz maps.
MSC 2000: Primary 30C65; Secondary 54B20.
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