Almost Injective Mappings of Totally Bounded Metric Spaces into Finite Dimensional Euclidean Spaces

Totally Bounded Metric Spaces

The papers [19], [9], [1], [4], [20], [2], [18], [13], [5], [8], [14], [21], [7], [15], [12], [11], [17], [6], [10], [16], and [3] provide the terminology and notation for this paper. For simplicity we follow the rules: M is a metric space, c, g are elements of the carrier of M , F is a family of subsets of the carrier of M , A, B are subsets of the carrier of M , f is a function, n, m, p, k ar...

Euclidean Quotients of Finite Metric Spaces

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α ≥ 1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embedings int...

Bilipschitz Embeddings of Metric Spaces into Euclidean Spaces

When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small (“snowflake”) deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, lik...

Approximation Algorithms for Bounded Dimensional Metric Spaces

The study of finite metrics is an important area of research, because of its wide applications to many different problems. The input of many problems (for instance clustering, near-neighbor queries and network routing) naturally involves a set of points on which a distance function has been defined. Hence, one would be motivated to store and process metrics in an efficient manner. The central i...

Euclidean Structure in Finite Dimensional Normed Spaces