The Fréchet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Fréchet matchings to “natural” matchings and to this...

A scalar valued set function on a Cartesian product of σ-algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequali...

The general question, “When is the product of Fréchet spaces Fréchet?” really depends on the questions of when a product of α4 Fréchet spaces (also known as strongly Fréchet or countably bisequential spaces) is α4, and when it is Fréchet. Two subclasses of the class of strongly Fréchet spaces shed much light on these questions. These are the class of α3 Fréchet spaces and its subclass of א0-bis...

The Fréchet mean generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, however, the Fréchet sample mean is not a consistent estimator of the theoretical Fréchet mean. For non-trivial examples, sequences of Fréchet sample mean sets may fail to converge in a set-analytical sense. Hence, it becomes necessary to consider other types of co...

The Fréchet distance is a natural measure of similarity between two curves [4]. The Fréchet distance between two curves is often referred to as the “dog-leash distance”. Alt and Godau [4] presented an algorithm to compute the Fréchet distance between two polygonal curves of n and m vertices in O(nm log(nm)) time. There has been a lot of applications using the Fréchet distance to do pattern/curv...

Given a curve f and a surface S, how hard is it to find a simple curve f ′ ⊆ S that is the most similar to f? We introduce and study this simple curve embedding problem for piecewise linear curves and surfaces in R and R, under Hausdorff distance, weak Fréchet distance, and Fréchet distance as similarity measures for curves. Surprisingly, while several variants of the problem turn out to have p...

The Fréchet mean or barycenter generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, the Fréchet sample mean is not a consistent estimator of the theoretical Fréchet mean. For graph-valued random variables, for instance, the Fréchet sample mean may fail to converge to a unique value. Hence, it becomes necessary to consider the converg...

Computing the Fréchet distance for surfaces is a surprisingly hard problem. It has been shown that it is NP-hard to compute the Fréchet distance between many nice classes of surfaces [God98], [Buc10]. On the other hand, a polynomial time algorithm exists for computing the Fréchet distance between simple polygons [Buc06]. This was the first paper to give an algorithm for computing the Fréchet di...

Introduction In a series of earlier papers, ([ST1-4]) we began a systematic study of locally analytic representations of a locally L-analytic group G, where L ⊆ C p is a finite extension of Q p. Such a representation is given by a continuous action of G on a locally convex topological vector space V over a spherically complete extension field K ⊆ C p of L, such that the orbit maps g → gv are lo...