We consider the problem of computing the time-convex hull of a point set under the general Lp metric in the presence of a straight-line highway in the plane. The traveling speed along the highway is assumed to be faster than that off the highway, and the shortest time-path between a distant pair may involve traveling along the highway. The time-convex hull TCH(P ) of a point set P is the smalle...

This paper provides a framework in order to incorporate reliability issue as a sign of disruption in distribution systems and partial covering theory as a response to limitation in coverage radios and economical preferences, simultaneously into the traditional literatures of capacitated facility location problems. As a result we develop a bi-objective model based on the discrete scenarios for e...

Let X be a linear space. A p-norm on X is a real-valued function on X with 0 < p ≤ 1, satisfying the following conditions: (i) ‖x‖p ≥ 0 and ‖x‖p = 0⇔ x = 0, (ii) ‖αx‖p = |α|p‖x‖p, (iii) ‖x+ y‖p ≤ ‖x‖p +‖y‖p for all x, y ∈ X and all scalars α. The pair (X ,‖,‖p) is called a p-normed space. It is a metric linear space with a translation invariant metric dp defined by dp(x, y)= ‖x− y‖p for all x, ...

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...

The finite-sample risk of the k nearest neighbor classifier that uses a weighted Lpmetric as a measure of class similarity is examined. For a family of classification problems with smooth distributions in Rn, an asymptotic expansion for the risk is obtained in decreasing fractional powers of the reference sample size. An analysis of the leading expansion coefficients reveals that the optimal we...

Based on Stein’s method, we derive upper bounds for Poisson process approximation in the L1-Wasserstein metric d (p) 2 , which is based on a slightly adapted Lp-Wasserstein metric between point measures. For the case p = 1, this construction yields the metric d2 introduced in [Barbour, A. D. and Brown, T. C. (1992), Stochastic Process. Appl. 43(1), pp. 9–31], for which Poisson process approxima...