Geodesic
Existence of a Lipschitz selection of the Chebyshev-centre map
Sbornik. Mathematics, Tome 204 (2013) no. 5, pp. 641-660 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper is concerned with the existence of a Lipschitz selection for the operator $T_C$ (the Chebyshev-centre map) that assigns to any bounded subset $M$ of a Banach space $X$ the set $T_C(M)$ of its Chebyshev centres. It is proved that if the unit sphere $S(X)$ of $X$ has an exposed smooth point, then $T_C$ does not have a Lipschitz selection. It is also proved that if $X$ is finite dimensional the operator $T_C$ has a Lipschitz selection if and only if $X$ is polyhedral. The operator $T_C$ is also shown to have a Lipschitz selection in the space $\mathbf c_0(K)$ and $\mathbf c$-spaces. Bibliography: 4 titles.
Keywords: Chebyshev centre, Lipschitz selection, metric projection. 
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     title = {Existence of {a~Lipschitz} selection of the {Chebyshev-centre} map},
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Yu. Yu. Druzhinin. Existence of a Lipschitz selection of the Chebyshev-centre map. Sbornik. Mathematics, Tome 204 (2013) no. 5, pp. 641-660. http://geodesic.mathdoc.fr/item/SM_2013_204_5_a1/

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