Geodesic
The best asymmetric approximation in spaces of continuous functions
Izvestiya. Mathematics , Tome 70 (2006) no. 4, pp. 809-839

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We consider approximation by convex sets in the space of continuous maps from a compact topological space to a locally convex space with respect to certain asymmetric seminorms. We suggest new criteria for elements of least deviation, make a definition of strongly unique elements of least deviation and study the problems of characterization and existence of such elements. The most detailed study concerns the approximation with a sign-sensitive weight of real-valued continuous functions defined on a compact metric space or on a line segment by elements of the Chebyshev space. 
@article{IM2_2006_70_4_a6,
     author = {A. V. Pokrovskii},
     title = {The best asymmetric approximation in spaces of continuous functions},
     journal = {Izvestiya. Mathematics },
     pages = {809--839},
     publisher = {mathdoc},
     volume = {70},
     number = {4},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2006_70_4_a6/}
}
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A. V. Pokrovskii. The best asymmetric approximation in spaces of continuous functions. Izvestiya. Mathematics , Tome 70 (2006) no. 4, pp. 809-839. http://geodesic.mathdoc.fr/item/IM2_2006_70_4_a6/