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This paper deals with the containment problem under homothetics which has theminimal enclosing ball (MEB) problem as a prominent representative. We connectthe problem to results in classic convex geometry and introduce a new series ofradii, which we call core-radii. For the MEB problem, these radii have alreadybeen considered from a different point of view and sharp inequalities betweenthem are known. In this paper sharp inequalities between core-radii for generalcontainment under homothetics are obtained. Moreover, the presentedinequalities are used to derive sharp upper bounds on the size of core-sets forcontainment under homothetics. In the MEB case, this yields a tight (dimensionindependent) bound for the size of such core-sets. In the general case, we showthat there are core-sets of size linear in the dimension and that this boundstays sharp even if the container is required to be symmetric.

No dimension independent Core-Sets for Containment under Homothetics