共查询到20条相似文献,搜索用时 15 毫秒
1.
Mordechai Novick 《Discrete and Computational Geometry》2012,48(4):1058-1073
Given a family F of n pairwise disjoint compact convex sets in the plane with non-empty interiors, let T(k) denote the property that every subfamily of F of size k has a line transversal, and T the property that the entire family has a line transversal. We illustrate the applicability of allowable interval sequences to problems involving line transversals in the plane by proving two new results and generalizing three old ones. Two of the old results are Klee??s assertion that if F is totally separated then T(3) implies T, and the following variation of Hadwiger??s Transversal Theorem proved by Wenger and (independently) Tverberg: If F is ordered and each four sets of F have some transversal which respects the order on F, then there is a transversal for all of F which respects this order. The third old result (a consequence of an observation made by Kramer) and the first of the new results (which partially settles a 2008 conjecture of Eckhoff) deal with fractional transversals and share the following general form: If F has property T(k) and meets certain other conditions, then there exists a transversal of some m sets in F, with k<m<n. The second new result establishes a link between transversal properties and separation properties of certain families of convex sets. 相似文献
2.
Abstract. For a region X in the plane, we denote by area(X) the area of X and by ℓ (∂ (X)) the length of the boundary of X . Let S be a convex set in the plane, let n ≥ 2 be an integer, and let α
1
, α
2
, . . . ,α
n
be positive real numbers such that α
1
+α
2
+ ⋅ ⋅ ⋅ +α
n
=1 and 0< α
i
≤ 1/2 for all 1 ≤ i ≤ n . Then we shall show that S can be partitioned into n disjoint convex subsets T
1
, T
2
, . . . ,T
n
so that each T
i
satisfies the following three conditions: (i) area(T
i
)=α
i
× area(S) ; (ii) ℓ (T
i
∩ ∂ (S))= α
i
× ℓ (∂ (S)) ; and (iii) T
i
∩ ∂ (S) consists of exactly one continuous curve. 相似文献
3.
Finding Convex Sets Among Points in the Plane 总被引:1,自引:0,他引:1
Let g(n) denote the least value such that any g(n) points in the plane in general position contain the vertices of a convex n -gon. In 1935, Erdős and Szekeres showed that g(n) exists, and they obtained the bounds Chung and Graham have recently improved the upper bound by 1; the first improvement since the original Erdős—Szekeres paper.
We show that
<lsiheader>
<onlinepub>26 June, 1998
<editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt;
<pdfname>19n3p405.pdf
<pdfexist>yes
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Received January 1, 1997, and in revised form June 6, 1997. 相似文献
4.
Abstract. For a region X in the plane, we denote by area(X) the area of X and by ℓ (∂ (X)) the length of the boundary of X . Let S be a convex set in the plane, let n ≥ 2 be an integer, and let α
1
, α
2
, . . . ,α
n
be positive real numbers such that α
1
+α
2
+ ⋅ ⋅ ⋅ +α
n
=1 and 0< α
i
≤ 1/2 for all 1 ≤ i ≤ n . Then we shall show that S can be partitioned into n disjoint convex subsets T
1
, T
2
, . . . ,T
n
so that each T
i
satisfies the following three conditions: (i) area(T
i
)=α
i
× area(S) ; (ii) ℓ (T
i
∩ ∂ (S))= α
i
× ℓ (∂ (S)) ; and (iii) T
i
∩ ∂ (S) consists of exactly one continuous curve. 相似文献
5.
The paper studies the relation between the asymptotic valuesof the ratios area/length (F/L) and diameter/length (D/L) ofa sequence of convex sets expanding over the whole hyperbolicplane. It is known that F/L goes to a value between 0 and 1depending on the shape of the contour. In the paper, it is firstof all seen that D/L has limit value between 0 and 1/2 in strongcontrast with the euclidean situation in which the lower boundis 1/ (D/L = 1/ if and only if the convex set has constant width).Moreover, it is shown that, as the limit of D/L approaches 1/2,the possible limit values of F/L reduce. Examples of all possiblelimits F/L and D/L are given. 相似文献
6.
Dedicated to the memory of Paul Erdős
Suppose we have a finite collection of closed convex sets in the plane, (which without loss of generality we can take to be
polygons). Suppose further that among any four of them, some three have non-empty intersection. We show that 13 points are
sufficient to meet every one of the convex sets.
Received October 27, 1999/Revised April 11, 2000
RID="*"
ID="*" Supported by grant OTKA-T-029074.
RID="†"
ID="†" Supported by NSF grant DMS-99-70071, OTKA-T-020914 and OTKA-F-22234. 相似文献
7.
In this paper, we establish lower bounds for n-term approximations in the metric of L
2(I
2
) of characteristic functions of plane convex subsets of the square I
2
with respect to arbitrary orthogonal systems. It is shown that, as n, these bounds cannot decrease more rapidly than
. 相似文献
8.
Géza Tóth 《Combinatorica》2000,20(4):589-596
Let F{\cal{F}} denote a family of pairwise disjoint convex sets in the plane. F{\cal{F}} is said to be in convex position, if none of its members is contained in the convex hull of the union of the others. For any fixed k 3 5k\ge5, we give a linear upper bound on Pk(n)P_k(n), the maximum size of a family F{\cal{F}} with the property that any k members of F{\cal{F}} are in convex position, but no n are. 相似文献
9.
10.
Lukács and András posed the problem of showing the existence of a set of n−2 points in the interior of a convex n-gon so that the interior of every triangle determined by three vertices of the polygon contains a unique point of S. Such sets have been called pebble sets by De Loera, Peterson, and Su. We seek to characterize all such sets for any given convex polygon in the plane.
We first consider a certain class of pebble sets, called peripheral because they consist of points that lie close to the boundary
of the polygon. We characterize all peripheral pebble sets, and show that for regular polygons, these are the only ones. Though
we demonstrate examples of polygons where there are other pebble sets, we nevertheless provide a characterization of the kinds
of points that can be involved in non-peripheral pebble sets. We furthermore describe algorithms to find such points. 相似文献
11.
A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by and its size by co(D) (cc(D)). Gutin et al. (2008) conjectured that the sum of the sizes of all convex sets (connected convex sets) in D equals Θ(n · co(D)) (Θ(n · cc(D))) where n is the order of D. In this paper we exhibit a family of connected acyclic digraphs with and . We also show that the number of connected convex sets of order k in any connected acyclic digraph of order n is at least n − k + 1. This is a strengthening of a theorem of Gutin and Yeo. 相似文献
12.
If C
1 is the convex hull of the curve of a standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex
hulls of C
1 and several rotations of it and compute the mean of the length of their perimeter by elementary calculations. This can be
seen geometrically as a study of the exit time by a Brownian motion from certain polytopes having the unit circle as an inscribed
one. 相似文献
13.
Epstein D. B. A.; Marden A.; Markovic V. 《Proceedings London Mathematical Society》2006,92(3):624-654
We make a detailed study of the relation of a euclidean convexregion C Dome (). The dome is the relative boundary, in theupper halfspace model of hyperbolic space, of the hyperbolicconvex hull of the complement of . The first result is to provethat the nearest point retraction r: Dome () is 2-quasiconformal.The second is to establish precise estimates of the distortionof r near . 2000 Mathematics Subject Classification 30C75,30F40, 30F45, 30F60. 相似文献
14.
We characterize the class of those closed convex sets which have a barrier cone with a nonempty interior. As a consequence, we describe the set of those proper extended-real-valued functionals for which the domain of their Fenchel conjugate has a nonempty interior. As an application, we study the stability of the solution set of a semi-coercive variational inequality. 相似文献
15.
In a seminal paper from 1935, Erdős and Szekeres showed that for each n there exists a least value g(n) such that any subset of g(n) points in the plane in general position must always contain the vertices of a convex n -gon. In particular, they obtained the bounds which have stood unchanged since then. In this paper we remove the +1 from the upper bound for n ≥ 4 .
<lsiheader>
<onlinepub>26 June, 1998
<editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt;
<pdfname>19n3p367.pdf
<pdfexist>yes
<htmlexist>no
<htmlfexist>no
<texexist>yes
<sectionname>
</lsiheader>
Received January 1, 1997, and in revised form June 6, 1997. 相似文献
16.
17.
Let
S{\mathcal{S}}
be a set system of convex sets in ℝ
d
. Helly’s theorem states that if all sets in
S{\mathcal{S}}
have empty intersection, then there is a subset
S¢ ì S{\mathcal{S}}'\subset{\mathcal{S}}
of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in
S{\mathcal{S}}
are not convex or if
S{\mathcal{S}}
does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the
aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets.
These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small
ε>0, the contraction C
−ε
and the expansion C
ε
are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with nonempty intersection: 相似文献
18.
N. G. Moshchevitin 《Mathematical Notes》2005,78(3-4):592-596
19.
Let E be a compact subset of C. We prove that if E satisfies the following local Markov property: for each polynomial P,
where M, m, s are positive constants independent of P,
and
; then E is L-regular, i.e. regular in the sense of the potential theory. In particular, if
satisfies the global Markov inequality, then E is L-regular. We also prove that if
is an m-perfect set (there exists c > 0 such that, for all
and $r\in (0,1]$,
and
, then E is L-regular. Examples given by Siciak [20] show that the assumption that m < 2 cannot be omitted. 相似文献
20.
G. E. Ivanov 《Mathematical Notes》2006,79(1-2):55-78
In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game. 相似文献