Abstract
A set in a metric space is called a Čebyšev set if it contains a unique “nearest neighbour” to each point of the space. In this paper we introduce the concept of a monotone arc of convex sets and show that compact monotone arcs have the Čebyšev property in the hyperspace of compact strictly convex sets. In the hyperspace of compact convex sets only certain monotone arcs are Čebyšev ; these are characterized. Results are also obtained for affine segments and for noncompact monotone arcs.
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References
Benson R.V.: Euclidean Geometry and Convexity. McGraw Hill, New York (1966)
Bisztriczky T.: On separated families of convex bodies. Arch. Math. 54, 193–199 (1990)
Bogdewicz A., Dawson R., Moszyńska M.: Čebyšev sets in hyperspaces over a Minkowski space. Glas. Mat. 42, 57–67 (2007)
Bogdewicz A., Moszyńska M.: Čebyšev sets in the space of convex bodies. Rend. Circ. Mat. Palermo (2), Suppl. 77, 19–39 (2006)
Braess D.: Nonlinear Approximation Theory. Springer, Berlin (1986)
Dawson, R.: Some Čebyšev sets with dimension d + 1 in hyperspaces over R d, In: Dawson, R., Herburt, I., Moszyńska, M., Pronk, D. (eds.) Banach Center Publications, vol. 84, pp. 89–110 (2009)
Dawson R., Edelstein M.: Families of bodies with definite common supports. Geom. Ded. 33, 195–204 (1990)
Dawson R., Moszyńska M.: Čebyšev sets in hyperspaces. Can. J. Math. 61, 299–314 (2009)
Federer H.: Curvature measures. Trans. AMS 93, 418–481 (1959)
Holmes R.B.: Geometric Functional Analysis and its Applications. Springer, Berlin (1975)
Klee V.: Convexity of Chebyshev sets. Math. Ann. 142, 292–304 (1961)
Poénaru V.: WHAT IS...an Infinite Swindle?. Not. AMS 54, (2007)
Singer I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer, Berlin (1970)
Sloane, N.: The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/njas/sequences/ (accessed April 21, 2009)
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Funded by a NSERC Discovery grant.
An erratum to this article can be found at http://dx.doi.org/10.1007/s00022-010-0050-2
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Dawson, R.J.M. Monotone and Čebyšev arcs in hyperspaces. J. Geom. 98, 1–19 (2010). https://doi.org/10.1007/s00022-010-0044-0
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DOI: https://doi.org/10.1007/s00022-010-0044-0