scholarly journals The different ways of stabbing disjoint convex sets

1992 ◽  
Vol 7 (2) ◽  
pp. 197-206 ◽  
Author(s):  
M. Katchalski ◽  
T. Lewis ◽  
A. Liu
Keyword(s):  
Convex Sets ◽  
Disjoint Convex

scholarly journals On the VC-dimension of half-spaces with respect to convex sets

2021 ◽  
Vol vol. 23, no. 3 (Combinatorics) ◽  
Author(s):  
Nicolas Grelier ◽  
Saeed Gh. Ilchi ◽  
Tillmann Miltzow ◽  
Shakhar Smorodinsky
Keyword(s):  
Pairwise Disjoint ◽  
Convex Sets ◽  
Range Query ◽  
Set Cover ◽  
Vc Dimension ◽  
Set Cover Problem ◽  
Structure Problem ◽  
Cover Problem ◽  
Geometric Set Cover ◽  
Disjoint Convex

A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.

Separating Translates in the Plane: Combinatorial Bounds and an Algorithm

1997 ◽  
Vol 07 (06) ◽  
pp. 551-562
Author(s):  
Jurek Czyzowicz ◽  
Hazel Everett ◽  
Jean-Marc Robert
Keyword(s):  
Pairwise Disjoint ◽  
Convex Sets ◽  
Convex Set ◽  
Time Algorithm ◽  
The Other ◽  
Separation Problem ◽  
Disjoint Convex

Given a family of pairwise disjoint convex sets S in the plane, a set [Formula: see text] is separated from a subset X ⊆ S if there exists a line l such that [Formula: see text] lies on one side of l and the sets in X lie on the other side. In this paper, we establish two combinatorial bounds related to the separation problem for families of n pairwise disjoint translates of a convex set in the plane: 1) there exists a line which separates one translate from at least [Formula: see text] translates, for some constant [Formula: see text] that depends only on the shape of the translates and 2) there is a function [Formula: see text], defined only by the shape of [Formula: see text], such that there exists a line with orientation Θ or [Formula: see text] which separates one translate from at least [Formula: see text] translates, for any orientation Θ. We also present an O(n log (n + k) + k) time algorithm for finding a translate which can be separated from the maximum number of translates amongst families of n pairwise disjoint translates of convex k-gons in the plane.

Metrizability of Finite Dimensional Spaces with a Binary Convexity

1986 ◽  
Vol 38 (1) ◽  
pp. 1-18 ◽  
Author(s):  
M. Van de Vel
Keyword(s):  
Convex Hull ◽  
Weak Topology ◽  
Convex Sets ◽  
Nonempty Intersection ◽  
Finite Collection ◽  
Closed Sets ◽  
Finite Dimensional ◽  
The One ◽  
Image Position ◽  
Disjoint Convex

A convex structure consists of a set X, together with a collection of subsets of X, which is closed under intersection and under updirected union. The members of are called convex sets, and is a convexity on X. Fox A a subset of X, h (A) denotes the (convex) hull of A. If A is finite, then h(A) is called a polytope, is called a binary convexity if each finite collection of pairwise intersecting convex sets has a nonempty intersection. See [8], [21] for general references.If X is also equipped with a topology, then the corresponding weak topology is the one generated by the convex closed sets. It is usually assumed that at least all polytopes are closed. is called normal provided that for each two disjoint convex closed sets C, D there exist convex closed sets C′, D′, with

The combinatorial encoding of disjoint convex sets in the plane

COMBINATORICA ◽  
2008 ◽  
Vol 28 (1) ◽  
pp. 69-81 ◽  
Author(s):  
Jacob E. Goodman ◽  
Richard Pollack
Keyword(s):  
Convex Sets ◽  
Combinatorial Encoding ◽  
Disjoint Convex

On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

1990 ◽  
Vol 48 (1) ◽  
pp. 520-521
Author(s):  
Neng-Yu Zhang ◽  
Bruce F. McEwen ◽  
Joachim Frank
Keyword(s):  
Function Space ◽  
Convex Sets ◽  
Electron Tomography ◽  
Spinal Ganglion ◽  
Fourier Space ◽  
Missing Information ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
Energy Bound ◽  
Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

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