  We study a general smallest intersecting ball problem and its soft-margin
variant in high-dimensional Euclidean spaces, which only require the input
objects to be compact and convex. These two problems link and unify a series of
fundamental problems in computational geometry and machine learning, including
smallest enclosing ball, polytope distance, intersection radius, $\ell_1$-loss
support vector machine, $\ell_1$-loss support vector data description, and so
on. Two general approximation algorithms are presented respectively, and
implementation details are given for specific inputs of convex polytopes,
reduced polytopes, axis-aligned bounding boxes, balls, and ellipsoids. For most
of these inputs, our algorithms are the first results in high-dimensional
spaces, and also the first approximation methods. To achieve this, we develop a
novel framework for approximating zero-sum games in Euclidean Jordan algebra
systems, which may be useful in its own right.
