Sets In Euclidean Space Research Articles

A Sampling Theory for Compact Sets in Euclidean Space

We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in euclidean space. These conditions are sufficient to ensure the topological correctness of a reconstruction given by an offset of the sampling. Our approach also yields new stability results for medial axes, critical points, and critical values of distance functions.

Porosity and the Darboux Property of Fréchet Derivatives

We study a relation between the porosity of sets in Euclidean spaces and the Darboux property of (relative) Frechet derivatives.

Normal cone approximation and offset shape isotopy

This work addresses the problem of the approximation of the normals of the offsets of general compact sets in Euclidean spaces. It is proven that for general sampling conditions, it is possible to approximate the gradient vector field of the distance to general compact sets. These conditions involve the μ-reach of the compact set, a recently introduced notion of feature size. As a consequence, we provide a sampling condition that is sufficient to ensure the correctness up to isotopy of a reconstruction given by an offset of the sampling. We also provide a notion of normal cone to general compact sets that is stable under perturbation.

Doubling measures and nonquasisymmetric maps on Whitney modification sets in Euclidean spaces

Let $E$ be a closed set in $\mathbb{R}^n$ and $\mathcal{W}$ a Whitney decomposition of $\mathbb{R}^n\setminus E$. Choosing one point from the interior of each cube in $\mathcal{W}$ we obtain a set $F$ and then we say that the set $E\cup F$ is a Whitney modification of $E$. The Whitney modification of a measure $\mu$ on $\mathbb{R}^n$ to $E\cup F$ is a measure $\nu$ defined on $E\cup F$ by $\nu\equiv\mu$ on $E$ and by $\nu(\{x\})=\mu(I_x)$ for every $x\in F$, where $I_x\in\mathcal{W}$ is the cube containing the point $x$. We prove that a measure on $E\cup F$ is doubling if and only if it is the Whitney modification of a doubling measure on $\mathbb{R}^n$. As its application, we show that there are metric spaces $X,Y$ and a nonquasisymmetric homeomorphism $f$ of $X$ onto $Y$ such that a measure $\mu$ on $X$ is doubling if and only if its image $\mu\circ f^{-1}$ is doubling on $Y$.

EMBEDDINGS OF COMPLEX LINE SYSTEMS AND FINITE REFLECTION GROUPS

AbstractA star is a planar set of three lines through a common point in which the angle between each pair is 60∘. A set of lines through a point in which the angle between each pair of lines is 60 or 90∘ is star-closed if for every pair of its lines at 60∘ the set contains the third line of the star. In 1976 Cameron, Goethals, Seidel and Shult showed that the indecomposable star-closed sets in Euclidean space are the root systems of types An, Dn, E6, E7 and E8. This result was a key part of their determination of all graphs with least eigenvalue −2. Subsequently, Cvetković, Rowlinson and Simić determined all star-closed extensions of these line systems. We generalize this result on extensions of line systems to complex n-space equipped with a hermitian inner product. There is one further infinite family, and two exceptional types arising from Burkhardt and Mitchell’s complex reflection groups in dimensions five and six. The proof is a geometric version of Mitchell’s classification of complex reflection groups in dimensions greater than four.

On ephemeral continua

Among the fundamentals of continuum mechanics, a cardinal tenet is the perfect identifiability of material elements. Consequently, the key mathematical tools one summons are bijections, of various degree of regularity, between sets in Euclidean space. Here we explore how one can manage without that tenet and what might ensue from the attendant want. The image one evokes directly is that of a loose shifting throng of minute pebbles, a numerous assembly of streaming motes. Thus, directly, the issues are relevant to fast granular flows; however, most results bear also on the study of complex media generally, even on a deeper analysis of response of standard continua.

Killed Brownian motion and inequalities among solutions of the Schrödinger equation

We construct triplets of killed Brownian motions to obtain inequalities relating the solutions of the Schrödinger equation 1 2 Δ ψ − h ψ = 0 , with non-negative boundary conditions, on three interrelated compact sets in Euclidean space. These, in particular, include inequalities relating harmonic functions on the three compact convex sets and an inequality relating to solutions of the h -equilibrium potential equations.

A weak characterisation of the Delaunay triangulation

We consider a new construction, the weak Delaunay triangulation of a finite point set in a metric space, which contains as a subcomplex the traditional (strong) Delaunay triangulation. The two simplicial complexes turn out to be equal for point sets in Euclidean space, as well as in the (hemi)sphere, hyperbolic space, and certain other geometries. There are weighted and approximate versions of the weak and strong complexes in all these geometries, and we prove equality theorems in those cases also. On the other hand, for discrete metric spaces the weak and strong complexes are decidedly different. We give a short empirical demonstration that weak Delaunay complexes can lead to dramatically clean results in the problem of estimating the homology groups of a manifold represented by a finite point sample.

Curvature measures and fractals

Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the ‘classical’ concepts of curvature are not directly applicable to fractal sets. We try to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets F ⊆ R d (e.g. fractals), for which classical geometric characteristics such as curvatures or Euler characteristic are not available, we study these notions for their -parallel sets Fe := {x ∈ R d : inf y∈F k x − yk ≤ }, instead, expecting that their limiting behaviour as → 0 does provide information about the structure of the initial set F. In particular, we investigate the limiting behaviour of the total curvatures (or intrinsic volumes) Ck(Fe), k = 0, . . . , d, as well as weak limits of the corresponding curvature measures Ck(Fe, � ) as → 0. This leads to the notions of fractal curvature and fractal curvature measure, respectively. The well known Minkowski content appears in this concept as one of the fractal curvatures. For certain classes of self-similar sets, results on the existence of (averaged) fractal curvatures are presented. These limits can be calculated explicitly and are in a certain sense ‘invariants’ of the sets, which may help to distinguish and classify fractals. Based on these results also the fractal curvature measures of these sets are characterized. As a special case and a significant refinement of known results, a local characterization of the Minkowski content is given.

Coordinate neighborhoods of arcs and the approximation of maps into (almost) complex manifolds

We study the approximation of J-holomorphic maps continuous to the boundary from a domain in into an almost complex manifold by maps J- holomorphic to the boundary, giving partial results in the non-integrable case. For the integrable case, we study arcs in complex manifolds and establish the existence of neighborhoods biholomorphic to open sets in Euclidean space for several classes of arcs. As an application we obtain C k approximation of holomorphic maps continuous to the boundary into complex manifolds by maps holomorphic to the boundary, provided the boundary is nice enough.

The normal cycle of a compact definable set

An elementary construction of the normal cycle of a compact definable set in Euclidean space (and more generally of a compactly supported constructible function) is given. Here “definable” means definable in some o-minimal structure. The construction is based on the notion of support function and uses only basic o-minimal geometry.

The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology

Let ConvF (R) be the space of all non-empty closed convex sets in Euclidean space R endowed with the Fell topology. In this paper, we prove that ConvF (R) ≈ R × Q for every n > 1 whereas ConvF (R) ≈ R× I. Let Conv(X) be the set of all non-empty closed convex sets in a normed linear space X = (X, ‖·‖). We can consider various topologies on Conv(X). In the paper [6], the AR-property of the spaces Conv(X) with the Hausdorff metric topology, the Attouch-Wets topology, and the Wijsman topology has been studied. In this paper, we shall consider the Fell topology on Conv(X), which is generated by the sets of the form U− = {A ∈ Conv(X) | A ∩ U 6= ∅} and (X \K)+ = {A ∈ Conv(X) | A ⊂ X \K}, where U is open and K is compact in X. This topology is also defined on the set Conv∗(X) = Conv(X) ∪ {∅}. By ConvF (X) and ConvF (X), we denote the spaces Conv∗(X) and Conv(X) admitting the Fell topology. In case X is finite-dimensional (equivalently locally compact), ConvF (X) is a locally compact metrizable space and ConvF (X) is its Alexandorff onepoint compactification. It is easy to see that ConvF ((0, 1)) is homeomorphic to (≈) the triangle with two vertices removed, ∆ {(0, 0), (1, 1)}, where ∆ = {(x, y) ∈ I | x 6 y} ⊂ I. Since ConvF (R) ≈ ConvF ((0, 1)), we have ConvF (R) ≈ ∆ {(0, 0), (1, 1)} ≈ R× I, hence it follows that ConvF (R) ≈ ∆/{(0, 0), (1, 1)} ≈ (S × I)/({pt} × I), 1991 Mathematics Subject Classification. 54B20, 54D05, 54E45, 57N20.

Distinct Distances in Homogeneous Sets in Euclidean Space

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $\Omega(n^{.6091})$ distinct distances.

Runge approximation on convex sets implies the Oka property

We prove that the classical Oka property of a complex manifold Y, concerning the existence and homotopy classification of holomorphic mappings from Stein manifolds to Y, is equivalent to a Runge approximation property for holomorphic maps from compact convex sets in Euclidean spaces to Y .

The Apollonian Metric: The Comparison Property, Bilipschitz Mappings and Thick Sets

The Apollonian metric is a generalization of the hyperbolic metric to arbitrary open sets in Euclidean spaces. In this article we show that the Apollonian metric is comparable to the jG metric in the set G if and only if its complement is unbounded and thick in the sense of Vaisala, Vuorinen and Wallin [Thick sets and quasisymmetric maps, Nagoya Math. J. 135 (1994), 121–148]. These conditions are also equivalent to the following: there exists L > 1 such that all Euclidean Lbilipschitz mappings are Apollonian bilipschitz with uniformly bounded constant.

Antipodality properties of finite sets in Euclidean space

This is a survey of known results and still open problems on antipodal properties of finite sets in Euclidean space. The exposition follows historical lines and takes into consideration both metric and affine aspects.

Large games with only small players and strategy sets in Euclidean spaces

Large games with only small players and strategy sets in Euclidean spaces

A Harnack Inequality for Ordinary Differential Equations

is true for all x and y in [a, b] and for each u in H that is not identically zero. In other words, the ratios of the values of the nonzero functions in H at any two points of [a, b] all lie in the fixed interval [1/C, C] that is independent of the functions. (We will see in section 4 that C does, in general, depend on A and B as well as on a and b.) Alex Harnack (1851-1888) first introduced inequality (1.2) in his book Grundlagen der Theorie des Logarithmischen Potentials (Leipzig, 1887). In this book, the inequality was stated for positive harmonic functions, and Harnack used it to show that a bounded sequence of harmonic functions on an open connected set in Euclidean space contains a subsequence that converges uniformly on compacta to a harmonic function. This fact is useful, in turn, for establishing the existence of a harmonic function on an open set 2 that takes prescribed continuous data on the boundary 302 [6, pp. 2326](i.e., for solving the classical Dirichlet problem). Recall that a harmonic function on an open set i2 in R is a function u in C2 (2) that satisfies Laplace's equation:

On the expected volume of intersection of independent Wiener sausages and the asymptotic behaviour of some related integrals

Let K be a compact, non-polar set in Euclidean space R m ( m ⩾ 3 ) and let T K be the first hitting time of K by a Brownian motion. We obtain the leading asymptotic behaviour as t → ∞ of ∫ R m dx ( P x [ T K < t ] ) k , where k > 0 is a constant.

Computable operators on regular sets

AbstractFor regular sets in Euclidean space, previous work has identified twelve ‘basic’ computability notions to (pairs of) which many previous notions considered in literature were shown to be equivalent. With respect to those basic notions we now investigate on the computability of natural operations on regular sets: union, intersection, complement, convex hull, image, and pre‐image under suitable classes of functions. It turns out that only few of these notions are suitable in the sense of rendering all those operations uniformly computable. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)