Embeddings of finite metric spaces into Euclidean space have been studied in several contexts: The local theory of Banach spaces, the design of approximation algorithms, and graph theory. The emphasis is usually on embeddings with the least possible distortion. That is, one seeks an embedding that minimizes the bi-Lipschitz constant of the mapping. This question has also been asked for embeddin...

It is proved that if 0 < c < 0.72/« then for any «-point metric space (X, d), the metric space (X,dc) is isometrically embeddable into a Euclidean space. For 6-point metric space, c = j log2 | is the largest exponent that guarantees the existence of isometric embeddings into a Euclidean space. Such largest exponent is also determined for all «-point graphs with "truncated distance".

For a set P of n points in R, the Euclidean 2-center problem computes a pair of congruent disks of the minimal radius that cover P . We extend this to the (2, k)-center problem where we compute the minimal radius pair of congruent disks to cover n − k points of P . We present a randomized algorithm with O(nk log n) expected running time for the (2, k)-center problem. We also study the (p, k)-ce...

We detail how to use Newton's method for distortion-based curved $r$-adaption a discrete high-order metric field while matching target geometry. Specifically, we combine two terms: distortion measuring the deviation from metric; and penalty term boundary. For this combination, consider four ingredients. First, represent field, log-Euclidean interpolation on (straight-edged) mesh. Second, interp...

It is a simple consequence of the maximum principle that a superharmonic function u on Rn(i. e. ∆u ≤ 0) which is 1 near infinity is identically 1 on Rn (throughout this paper, n ≥ 3). Geometrically this means that one can not conformally deform the Euclidean metric in a bounded region without decreasing the scalar curvature somewhere. In fact there is a much stronger result: one can not have an...