The Intrinsic Measure Theory of Riemannian and Euclidean Metric Spaces

1944 ◽  
Vol 45 (2) ◽  
pp. 367 ◽  
Author(s):  
Lynn H. Loomis
Keyword(s):  
Metric Spaces ◽  
Measure Theory ◽  
Euclidean Metric ◽  
Intrinsic Measure

scholarly journals Improved Performance of Asymptotically Optimal Rapidly Exploring Random Trees

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Beth Boardman ◽  
Troy Harden ◽  
Sonia Martínez
Keyword(s):  
Degrees Of Freedom ◽  
Metric Spaces ◽  
Dynamic Environments ◽  
Random Trees ◽  
Asymptotically Optimal ◽  
Euclidean Metric ◽  
The Third ◽  
Dubins Vehicle ◽  
Improved Performance ◽  
Dubins Vehicles

Three algorithms that improve the performance of the asymptotically optimal Rapidly exploring Random Tree (RRT*) are presented in this paper. First, we introduce the Goal Tree (GT) algorithm for motion planning in dynamic environments where unexpected obstacles appear sporadically. The GT reuses the previous RRT* by pruning the affected area and then extending the tree by drawing samples from a shadow set. The shadow is the subset of the free configuration space containing all configurations that have geodesics ending at the goal and are in conflict with the new obstacle. Smaller, well defined, sampling regions are considered for Euclidean metric spaces and Dubins' vehicles. Next, the Focused-Refinement (FR) algorithm, which samples with some probability around the first path found by an RRT*, is defined. The third improvement is the Grandparent-Connection (GP) algorithm, which attempts to connect an added vertex directly to its grandparent vertex instead of parent. The GT and GP algorithms are both proven to be asymptotically optimal. Finally, the three algorithms are simulated and compared for a Euclidean metric robot, a Dubins' vehicle, and a seven degrees-of-freedom manipulator.

scholarly journals Weak Convergence of Topological Measures

2021 ◽  
Author(s):  
Svetlana V. Butler
Keyword(s):  
Probability Theory ◽  
Weak Convergence ◽  
Metric Spaces ◽  
Measure Theory ◽  
Convergent Subsequence ◽  
Borel Measures ◽  
Nonlinear Functionals ◽  
Topological Measures ◽  
Standard Techniques ◽  
Uniformly Bounded

AbstractTopological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (nonlinear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov’s theorem for equivalent definitions of weak convergence of deficient topological measures. We also prove a version of Prokhorov’s theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich–Rubenstein metrics and show that convergence in either of them implies weak convergence of (deficient) topological measures on metric spaces. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures. The present paper constitutes a necessary step to further research in probability theory and its applications in the context of (deficient) topological measures and corresponding nonlinear functionals.

scholarly journals Union of Euclidean Metric Spaces is Euclidean

2016 ◽  
pp. 1-14 ◽  
Author(s):  
Konstantin Makarychev ◽  
Yury Makarychev
Keyword(s):  
Metric Spaces ◽  
Euclidean Metric

RECTIFIABILITY OF MEASURES WITH LOCALLY UNIFORM CUBE DENSITY

2003 ◽  
Vol 86 (1) ◽  
pp. 153-249 ◽  
Author(s):  
ANDREW LORENT
Keyword(s):  
Euclidean Space ◽  
Isometric Immersion ◽  
Metric Spaces ◽  
Model Problem ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Initial Condition ◽  
Radon Measures ◽  
Geometric Measure ◽  
Euclidean Unit Ball

The conjecture that Radon measures in Euclidean space with positive finite density are rectifiable was a central problem in Geometric Measure Theory for fifty years. This conjecture was positively resolved by Preiss in 1986, using methods entirely dependent on the symmetry of the Euclidean unit ball. Since then, due to reasons of isometric immersion of metric spaces into $l_{\infty}$ and the uncommon nature of the sup norm even in finite dimensions, a popular model problem for generalising this result to non-Euclidean spaces has been the study of 2-uniform measures in $l^{3}_{\infty}$. The rectifiability or otherwise of these measures has been a well-known question.In this paper the stronger result that locally 2-uniform measures in $l^{3}_{\infty}$ are rectifiable is proved. This is the first result that proves rectifiability, from an initial condition about densities, for general Radon measures of dimension greater than 1 outside Euclidean space.2000 Mathematical Subject Classification: 28A75.

scholarly journals On the Plane Geometry with Generalized Absolute Value Metric

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
A. Bayar ◽  
S. Ekmekçi ◽  
Z. Akça
Keyword(s):  
Metric Spaces ◽  
Plane Geometry ◽  
Trigonometric Functions ◽  
Absolute Value ◽  
Euclidean Metric

Metric spaces are among the most important widely studied topics in mathematics. In recent years, Mathematicians began to investigate using other metrics different from Euclidean metric. These metrics also find their place computer age in addition to their importance in geometry. In this paper, we consider the plane geometry with the generalized absolute value metric and define trigonometric functions and norm and then give a plane tiling example for engineers underlying Schwarz's inequality in this plane.

Pointless metric spaces

1990 ◽  
Vol 55 (1) ◽  
pp. 207-219 ◽  
Author(s):  
Giangiacomo Gerla
Keyword(s):  
Lower Bound ◽  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Space Theory ◽  
Research Projects ◽  
Natural Numbers ◽  
Real Numbers ◽  
Euclidean Metric

In the last years several research projects have been motivated by the problem of constructing the usual geometrical spaces by admitting “regions” and “inclusion” between regions as primitives and by defining the points as suitable sequences or classes of regions (for references see [2]).In this paper we propose and examine a system of axioms for the pointless space theory in which “regions”, “inclusion”, “distance” and “diameter” are assumed as primitives and the concept of point is derived. Such a system extends a system proposed by K. Weihrauch and U. Schreiber in [5].In the sequel R and N denote the set of real numbers and the set of natural numbers, and E is a Euclidean metric space. Moreover, if X is a subset of R, then ⋁X is the least upper bound and ⋀X the greatest lower bound of X.

scholarly journals Some fine properties of currents and applications to distributional Jacobians

2002 ◽  
Vol 132 (4) ◽  
pp. 815-842 ◽  
Author(s):  
Camillo De Lellis
Keyword(s):  
Metric Spaces ◽  
Special Functions ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Finite Mass ◽  
Geometric Measure ◽  
Closure Theorem

We study fine properties of currents in the framework of geometric measure theory on metric spaces developed by Ambrosio and Kirchheim, and we prove a rectifiability criterion for flat currents of finite mass. We apply these tools to study the structure of the distributional Jacobians of functions in the space BnV, defined by Jerrard and Soner. We define the subspace of special functions of bounded higher variation and we prove a closure theorem.

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