Showing papers on "Orthogonal convex hull published in 1999"

Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization

Abstract: The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility problems, linear inequalities, for instance). It is shown that the standard constraint qualification from convex analysis implies bounded linear regularity, which in turn yields the strong conical hull intersection property. Jameson's duality for two cones, which relates bounded linear regularity to property (G), is re-derived and refined. For polyhedral cones, a statement dual to Hoffman's error bound result is obtained. A sharpening of a result on error bounds for convex inequalities by Auslender and Crouzeix is presented. Finally, for two subspaces, property (G) is quantified by the angle between the subspaces.

A geometric interpretation of ν-SVM classifiers

Abstract: We show that the recently proposed variant of the Support Vector machine (SVM) algorithm, known as ν-SVM, can be interpreted as a maximal separation between subsets of the convex hulls of the data, which we call soft convex hulls. The soft convex hulls are controlled by choice of the parameter ν. If the intersection of the convex hulls is empty, the hyperplane is positioned halfway between them such that the distance between convex hulls, measured along the normal, is maximized; and if it is not, the hyperplane's normal is similarly determined by the soft convex hulls, but its position (perpendicular distance from the origin) is adjusted to minimize the error sum. The proposed geometric interpretation of ν-SVM also leads to necessary and sufficient conditions for the existence of a choice of ν for which the ν-SVM solution is nontrivial.

Geometry of self-affine tiles ii

Abstract: We continue the study in part I of geometric properties of self similar and self a ne tiles We give some experimental results from implementing the algorithm in part I for computing the dimension of the boundary of a self similar tile and we describe some conjectures that result We prove that the dimension of the boundary may assume values arbitrarily close to the dimension of the tile We give a formula for the area of the convex hull of a planar self a ne tile We prove that the extreme points of the convex hull form a set of dimension zero and we describe a natural gauge function for this set Introduction to Part II This paper is a continuation of SW which we refer to as part I and the sections are numbered accordingly In Section of part I we obtained an algorithm for computing the dimension box and Hausdor dimensions are equal of the boundary in the case of a self similar tile satisfying

H-Walk: hierarchical distance computation for moving convex bodies

Abstract: This paper presents the Hierarchical Walk, or H-Walk algorithm, which maintains the distance between two moving convex bodies by exploiting both motion coherence and hierarchical representations. For convex polygons, we prove that H-Walk improves on the classic Lin-Canny and DobkinKirkpatrick algorithms. We have implemented H-Walk for moving convex polyhedra in three dimensions. Experimental results indicate that, unlike previous incremental distance computation algorithms, H-Walk adapts well to variable coherence in the motion.

Limit theorems for the convex hull of random points in higher dimensions

Abstract: We give a central limit theorem for the number Nn of vertices of the convex hull of n independent and identically distributed random vectors, being sampled from a certain class of spherically symmetric distributions in Rd (d > 1), that includes the normal family. Furthermore, we prove that, among these distributions, the variance of Nn exhibits the same order of magnitude as the expectation as n→∞. The main tools are Poisson approximation of the point process of vertices of the convex hull and (sub/super)-martingales.

Sylvester's question : The probability that n points are in convex position

Abstract: For a convex body $K$ in the plane, let $p(n, K)$ denote the probability that $n$ random, independent, and uniform points from $K$ are in convex position, that is, none of them lies in the convex hull of the others. Here we determine the asymptotic behavior of $p(n, K)$ by showing that, as $n$ goes to infinity,$ {n^2}^n\sqrt{p(n,K)}$ tends to a finite and positive limit.

On the Boundary of the Union of Planar Convex Sets

Abstract: Keywords: planar convex sets ; boundaries ; regular and irregular vertices ; arcs ; inequalities Note: Professor Pach's number: [126] Reference DCG-ARTICLE-1999-002doi:10.1007/PL00009424 Record created on 2008-11-14, modified on 2017-05-12

An exact minimum zone solution for three-dimensional straightness evaluation problems

Abstract: A mathematical model giving the exact solution for three-dimensional (3-D) straightness evaluation based on the minimum zone criterion is presented in this paper. This model builds upon the convex hull of the measured point set to show that the 3-D straightness is determined simply by some measured points on the vertices of the convex hull. The axis of the critical cylinder giving the minimum objective value is proved to be parallel to one of the edges of the convex hull. This model is effective and easy for implementation.

Distance to the convex hull of an orbit under the action of a compact Lie group

Abstract: For a real vector space V acted on by a group K and fixed x and y in V , we consider the problem of finding the minimum (resp., maximum) distance, relative to a Kinvariant convex function on V , between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V . Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component p of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

Optimal line bipartitions of point sets

Abstract: Let S be a set of n points in the plane. We study the following problem: Partition S by a line into two subsets Sa and Sb such that max{f(Sa), f(Sb)} is minimal, where f is any monotone function defined over 2S. We first present a solution to the case where the points in S are the vertices of a convex polygon and apply it to some common cases — f(S′) is the perimeter, area, or width of the convex hull of S′⊆S — to obtain linear solutions (or O(n log n) solutions if the convex hull of S is not given) to the corresponding problems. This solution is based on an efficient procedure for finding a minimal entry in matrices of some special type, which we believe is of independent interest. For the general case we present a linear space solution which is in some sense output sensitive. It yields solutions to the perimeter and area cases that are never slower and often faster than the best previous solutions.

Convex Hull of Grid Points below a Line or a Convex Curve

Abstract: onsider a finite non-vertical, and non-degenerate straight-line segment s =[s0 s1] in the Euclidian plane E2. We give a method for constructing the boundary of the upper convex hull of all the points with integral coordinates below (or on) s, with abscissa in [x(s0),x(s1)]. The algorithm takes O(log n) time, if n is the length of the segment. We next show how to perform a similar construction in the case where s is a finite, non-degenerate, convex arc on a quadric curve. The associated method runs in O(klog n), where n is the arc's length and k the number of vertices on the boundary of the resulting hull. This method may also be used for a line segment; in this case, k = O(log n), and the second method takes O(k2) time, compared with O(k) for the first.

On visibility and covering by convex sets

Abstract: A setX⊆ℝd isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn6 convex sets We improve their bound to 18n3, and show a lower bound of order Ω(n2) We also show that ifX⊆ℝ2 is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n4+n2λ) convex sets Two other results onn-convex sets are stated in the introduction (Corollary 12 and Proposition 14)

Partitioning point sets in space into disjoint convex polytopes

Abstract: We study the problem of partitioning point sets in the space so that each equivalence class is a convex polytope disjoint from the others. For a set of n points P in R 3 , define f(P) to be the minimum number of sets in a partition into disjoint convex polytopes of P and F(n) as the maximum of f(P), over all sets P of n points. We show that ⌈n/2(log2n+1)⌉≤F(n)≤⌈2n/9⌉. The lower bound also holds for partition into empty convex polytopes.

Convex hull of a planar set of straight and circular line segments

Abstract: The problem of constructing the convex hull of a set of points and of curvilinear segments arises in many applications of geometric analysis. Although there has been much work on algorithms for the convex hull of a finite point set, there has been less on methods for dealing with circular line segments and the implementation issues. This paper describes a new method for finding the convex hull of a planar set of straight and circular line segments. It then concentrates on the implementation of the algorithm.

Accelerating algorithm for 3D convex hulls construction

Abstract: The convex hull for a set of points is consist only of a small part of the set generally. If we can remove those points lying in the interior of the hull in an efficient way,we can then construct the hull with fewer time. By dividing the bounding box of the point set and by exploiting the convex property of a convex set,we get an efficient quick hull algorithm. The new method is simple to implement and more efficient than some traditional accelerating methods.

Circumradius-diameter and width-inradius relations for lattice constrained convex sets

Abstract: Let K be a planar, compact, convex set with circumradius R , diameter d , width w and inradius r , and containing no points of the integer lattice. We generalise inequalities concerning the ‘dual’ quantities (2 R − d ) and ( w − 2 r ) to rectangular lattices. We then use these results to obtain corresponding inequalities for a planar convex set with two interior lattice points. Finally, we conjecture corresponding results for sets containing one interior lattice point.

A parallel algorithm for convex hull of planar disperse point set

Abstract: A parallel algorithm for finding the convex hull of a planar discrete point set is put forward in this paper.The time for product operation is \$O(\%log\%\+3n)\$.It improves the traditional methods greatly.

Closed convex hull of the family of multivalently close-to-convex functions of order β

Abstract: In this paper the closed convex hulls of the compact familiesCβ(p), of multivalently close to convex functions of order β andV0k(p), of multivalent functions of bounded boundary rotation, have been determined, respectively for β≥1 andk≥2(p+1)/p. Extreme points of these convex hulls are partially characterised. For a fixed pointz0∈D={z:|z| 0. It is shown that iff is subordinate to some function inCβ(p) then each Taylor coefficient off is dominated by the corresponding coefficient of the function\(\smallint _0^z pt^{p - 1} (1 + t)^\beta /(1 - t)^{2p + \beta } dt\).

Simple Fast Convex Hull Algorithm of Planar Point Set

Abstract: Convex hull problem is one of the fundamental problems incomputational geometry,and used in many fields. The traditional convex hull algorithms of point set and that of simple polygons have parallelly developed without any combination. In this paper, a new algorithm is proposed based on combining the convex hull algorithms of simple polygons into solving the problem of point set. The algorithms in this paper not only reaches the lower bound of O( n log n ),but also is very simple and easy to be realized. The presented algorithms has been applied in plantdesign system of PDSOFT.The results obtained by the method areremarkable.