Geodesic
Uniform estimate of a~compact convex set by a~ball in an~arbitrary norm
Sbornik. Mathematics, Tome 191 (2000) no. 10, pp. 1433-1458

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The problem of the best uniform approximation of a compact convex set by a ball with respect to an arbitrary norm in the Hausdorff metric corresponding to that norm is considered. The question is reduced to a convex programming problem, which can be studied by means of convex analysis. Necessary and sufficient conditions for the solubility of this problem are obtained and several properties of its solution are described. It is proved, in particular, that the centre of at least one ball of best approximation lies in the compact set under consideration; in addition, conditions ensuring that the centres of all balls of best approximation lie in this compact set and a condition for unique solubility are obtained. 
@article{SM_2000_191_10_a1,
     author = {S. I. Dudov and I. V. Zlatorunskaya},
     title = {Uniform estimate of a~compact convex set by a~ball in an~arbitrary norm},
     journal = {Sbornik. Mathematics},
     pages = {1433--1458},
     publisher = {mathdoc},
     volume = {191},
     number = {10},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_10_a1/}
}
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S. I. Dudov; I. V. Zlatorunskaya. Uniform estimate of a~compact convex set by a~ball in an~arbitrary norm. Sbornik. Mathematics, Tome 191 (2000) no. 10, pp. 1433-1458. http://geodesic.mathdoc.fr/item/SM_2000_191_10_a1/