Another geometric proof of supercompactness of compact metric spaces

Compact Quantum Metric Spaces

We give a brief survey of many of the high-lights of our present understanding of the young subject of quantum metric spaces, and of quantum Gromov-Hausdorff distance between them. We include examples. My interest in developing the theory of compact quantum metric spaces was stimulated by certain statements in the high-energy physics and string-theory literature, concerning non-commutative spac...

Geometric embeddings of metric spaces

Another proof of Banaschewski's surjection theorem

We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform subl...

Covering compact metric spaces greedily

Abstract. A general greedy approach to construct coverings of compact metric spaces by metric balls is given and analyzed. The analysis is a continuous version of Chvátal’s analysis of the greedy algorithm for the weighted set cover problem. The approach is demonstrated in an exemplary manner to construct efficient coverings of the n-dimensional sphere and n-dimensional Euclidean space to give ...

Learning Over Compact Metric Spaces

We consider the problem of learning on a compact metric space X in a functional analytic framework. For a dense subalgebra of Lip(X), the space of all Lipschitz functions on X, the Representer Theorem is derived. We obtain exact solutions in the case of least square minimization and regularization and suggest an approximate solution for the Lipschitz classifier.