New results on q-positivity

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q-positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.Comment: 18 page

Two-Sided Kirszbraun Theorem

In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ?^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ? f from Y to ?^m. While the extension ? f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ? f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ?)-Lipschitz outer extension f?:Y ? ?^{m\u27} that does not decrease distances more than "necessary". Namely, ?f?(x) - f?(y)? ? c ?{?} min(?x-y?, inf_{a,b ? X} (?x - a? + ?f(a) - f(b)? + ?b-y?)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ?g(x) - g(y)? > L min(?x-y?, inf_{a,b ? X} (?x - a? + ?f(a) - f(b)? + ?b-y?)) even for a single pair of points x and y. In some applications, one is interested in the distances ?f?(x) - f?(y)? between images of points x,y ? Y rather than in the map f? itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ? f first. In contrast, our theorem provides a simple approximate formula for distances ?f?(x) - f?(y)?

Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph

Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric. In fact, we prove the surprising result that a previously published $3$-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest.Comment: 15 page

Alexandrov geometry: preliminary version no. 1

This is a preliminary version of our book. It goes up to the definition of dimension, which is about 30% of the material we plan to include. If you use it as a reference, do not forget to include the version number since the numbering will be changed.Comment: 238 pages, 35 figure