Relations between several problems of estimating convex compacta by balls

Finite-dimensional problems of finding outer, inner, and uniform estimates for a convex compactum by a ball in an arbitrary norm are considered and compared, as well as the problem of finding estimates of the boundary of a convex compactum by a spherical annulus of the smallest width. It is shown that these problems can be linked by means of the parametric problem of finding the best approximation in the Hausdorff metric of the compactum under consideration by a ball of fixed radius. One can indicate ranges of the fixed radius in which solutions of the latter problem give solutions of the problems mentioned above. However, for some values of the radius this latter problem can be independent.
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