Geodesic
Best Approximations of Convex Compact Sets by Balls in the Hausdorff Metric
Matematičeskie zametki, Tome 76 (2004) no. 2, pp. 226-236.

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We deduce an upper bound for the Hausdorff distance between a nonempty bounded set and the set of all closed balls in a strictly convex straight geodesic space $X$ of nonnegative curvature. We prove that the set $\chi[M]$ of centers of closed balls approximating a convex compact set in the Hausdorff metric in the best possible way is nonempty $X[M]$ and is contained in $M$. Some other properties of $\chi[M]$ also are investigated. 
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E. N. Sosov. Best Approximations of Convex Compact Sets by Balls in the Hausdorff Metric. Matematičeskie zametki, Tome 76 (2004) no. 2, pp. 226-236. http://geodesic.mathdoc.fr/item/MZM_2004_76_2_a6/

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