How Quasiconvexity works part1(Machine Learning)

<p>Let X be a metric space and BCl(X) the collection of its bounded closed subsets as a metric space with respect to Hausdorff distance (and call BCl(X) the bounded-subset space of X). The question of whether or not one can characterize (the existence of) a rectifiable path in some subspace J of BCl(X) entirely in terms of rectifiable paths in X does not seem to have been given serious consideration. In this paper, we make some progress with the case where J consists of precompact subsets of X (with such a J called a precompact-subset space of X). Specifically, in certain precompact-subset spaces J of X, we give a criterion to determine (the existence of) a rectifiable path in J using rectifiable paths in X. We then show that certain path connectivity properties, especially quasiconvexity, inherited from X by such precompact-subset spaces of X can be determined in an automatic way using our criterion. Meanwhile, we also give a concise review of our earlier work on quasiconvexity of finite-subset spaces of X</p> <p><a href="https://medium.com/@monocosmo77/how-quasiconvexity-works-part1-machine-learning-d0896dddaf7e">Website</a></p>