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1.
Oleg Topală 《Journal of Geometry》2001,70(1-2):164-167
Let S be a simply connected orthogonal polygon in and let P(S) denote the intersection of all maximal starshaped via staircase paths orthogonal subpolygons in S. Our result: if , then there exists a maximal starshaped via staircase paths orthogonal polygon , such that . As a corollary, P(S) is a starshaped (via staircase paths) orthogonal polygon or empty. The results fail without the requirement that the set
S is simply connected.
Received 1 March 1999. 相似文献
2.
Marilyn Breen 《Periodica Mathematica Hungarica》2007,55(2):169-176
A Krasnosel’skii-type theorem for compact sets that are starshaped via staircase paths may be extended to compact sets that
are starshaped via orthogonally convex paths: Let S be a nonempty compact planar set having connected complement. If every
two points of S are visible via orthogonally convex paths from a common point of S, then S is starshaped via orthogonally convex paths. Moreover, the associated kernel Ker S has the expected property that every two of its points are joined in Ker S by an orthogonally convex path. If S is an arbitrary nonempty planar set that is starshaped via orthogonally convex paths, then for each component C of Ker S, every two of points of C are joined in C by an orthogonally convex path.
Communicated by Imre Bárány 相似文献
3.
Marilyn Breen 《Archiv der Mathematik》1997,68(1):60-64
Let S be a nonempty closed, simply connected set in the plane, and let α τ; 0. If every three points of 5 see a common point of
S via paths of length at most α, then for some point s0 of S, s0 sees each point of S via such a path. That is, S is starshaped via paths of length at most α.
Supported in part by NSF grant DMS-9207019 相似文献
4.
Marilyn Breen 《Geometriae Dedicata》1994,53(1):49-56
LetT be a simply connected orthogonal polygon having the property that for every three points ofT, at least two of these points see each other via staircases inT. ThenT is a union of three orthogonally convex polygons. The number three is best possible.ForT a simply connected orthogonal polygon,T is a union of two orthogonally convex polygons if and only if for every sequencev
1,...,v
n,v
n+1 =v
1 inT, n odd, at least one consecutive pairv
i
,v
i+1 sees each other via staircase paths inT, 1 i n. An analogous result says thatT is a union of two orthogonal polygons which are starshaped via staircase paths if and only if for every odd sequence inT, at least one consecutive pair sees a common point via staircases inT.Supported in part by NSF grants DMS-8908717 and DMS-9207019. 相似文献
5.
Marilyn Breen 《Archiv der Mathematik》2003,80(6):664-672
Let $\cal{F}$ be a finite family of simply connected
orthogonal polygons in the plane. If every three (not necessarily
distinct) members of $\cal{F}$ have a nonempty intersection which
is starshaped via staircase paths, then the
intersection $\cap \{F : F\; \hbox{in}\; \cal{F}\}$
is a (nonempty) simply connected orthogonal polygon which is starshaped
via staircase paths. Moreover, the number three is best possible, even
with the additional requirement that the intersection in question be
nonempty. The result fails without the simple connectedness condition. 相似文献
6.
Marilyn Breen 《Journal of Geometry》2001,71(1-2):26-33
Let S be a simply connected orthogonal polygon in the plane, and assume that S is two-guardable (but not starshaped) via staircase paths. If K is a component of two-kernel (S), then the set of partners of points in K determines a second component K' of two-kernel (S). Thus the components occur in pairs. Moreover, each component is geodesically convex. The results fail without the requirement
that S be simply connected.
Received 2 November 1999; revised 28 September 2000. 相似文献
7.
Marilyn Breen 《Monatshefte für Mathematik》2003,139(2):105-109
Let S be a nonempty closed, simply connected set in the plane. For α > 0, let ℳ denote the family of all maximal subsets of S which are starshaped via paths of length at most α. Then ⋂{M : M in ℳ} is either starshaped via α-paths or empty. The result fails without the simple connectedness condition. However, even
with a simple connectedness requirement, there is no Helly theorem for intersections of sets which are starshaped via α-paths.
Received November 19, 2001; in revised form April 25, 2002
Published online November 18, 2002 相似文献
8.
Let S be a subset of R
d
. The set S is said to be an set if and only if for every two points x and y of S, there exists some z S such that [x, z] [z, y] S. Clearly every starshaped set is an set, yet the converse is false and introduces an interesting question: Under what conditions will an set S be almost starshaped; that is, when will there exist a convex subset C of S such that every point of S sees some point of C via SThis paper provides one answer to the question above, and we have the following result: Let S be a closed planar set, S simply connected, and assume that the set Q of points of local nonconvexity of S is finite. If some point p of S see each member of Q via S, then there is a convex subset C of S such that every point of S sees some point of C via S. 相似文献
9.
Victor Chepoi 《Geometriae Dedicata》1996,63(3):321-329
Let P
n
be a union of a finite number of boxes whose intersection graph is a tree. If every two boundary points of P are visible via staircase paths from a common point of P, then P is starshaped via staircase paths. The same result holds true when P is a cubical polyhedron of
n
, which is the geometric realization of some median graph.This generalizes the recent result of M. Breen, J. Geometry, 51 (1994), established for simple rectilinear polygons.Research for this paper was done while the author was visiting the Mathematisches Seminar der Universität Hamburg, on leave from the Universitatea de Stat din Moldova. The author gratefully acknowledges financial support by the Alexander von Humboldt Stifting. 相似文献
10.
Marilyn Breen 《Journal of Geometry》1989,36(1-2):8-16
LetS be a compact, simply connected set inR
2. If every boundary point ofS is clearly visible viaS from at least one of the three pointsa, b, c, thenS is a union of three starshaped sets whose kernels containa, b, c, respectively. The result fails when the number three is replaced by four.As a partial converse, ifS is a union of three starshaped sets whose kernels containa, b, c, respectively, then the set of points in the boundary ofS clearly visible from at least one ofa, b, orc is dense in the boundary ofS.Supported in part by NSF grant DMS-8705336. 相似文献
11.
A compact set is staircase connected if every two points a, b ∈ S can be connected by an x-monotone and y-monotone polygonal path with sides parallel to the coordinate axes. In [5] we have introduced the concepts of staircase k-stars and kernels.
In this paper we prove that if the staircase k-kernel is not empty, then it can be expressed as the intersection of a covering family of maximal subsets of staircase diameter
k of S.
相似文献
12.
Marilyn Breen 《Journal of Geometry》1989,35(1-2):14-18
SetS inR
d has propertyK
2 if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c1,c2 (depending onF) such that each point ofF is clearly visible viaS from at least one ci,i = 1,2. The following characterization theorem is established: Let
, d2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S
j
} converging toS (relative to the Hausdorff metric) such that each setS
j satisfies propertyK
2. For
, the sufficiency of the condition above still holds, although the necessity fails. 相似文献
13.
Marilyn Breen 《Monatshefte für Mathematik》2006,148(2):91-100
For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For
any finite family of simply connected orthogonal polygons in the plane and points x and y in
, if every r (not necessarily distinct) members of
contain a common staircase n-path from x to y, then
contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths.
Moreover, we establish the following dual result for unions of these sets: Let
be any finite family of orthogonal polygons in the plane, with
simply connected. If every three (not necessarily distinct) members of
have a union which is starshaped via staircase n-paths, then T is starshaped via staircase (n + 1)-paths. The number n + 1 in the theorem is best for every n ≥ 2. 相似文献
14.
Marilyn Breen 《Geometriae Dedicata》1982,13(2):201-213
Let Sø be a bounded connected set in R
2, and assume that every 3 or fewer lnc points of S are clearly visible from a common point of S. Then for some point p in S, the set A{s : s in S and [p, s] S} is nowhere dense in S. Furthermore, when S is open, then S in starshaped. 相似文献
15.
Marilyn Breen 《Journal of Geometry》1999,65(1-2):50-53
Let
be a finite family of compact sets in the plane, and letk be a fixed natural number. If every three (not necessarily distinct) members of
have a union which is simply connected and starshaped viak-paths, then
and
is starshaped viak-paths. Analogous results hold for paths of length at most , > 0, and for staircase paths, although not for staircasek-paths.Supported in part by NSF grant DMS-9504249 相似文献
16.
Marilyn Breen 《Aequationes Mathematicae》2010,79(1-2):99-110
Let S be a simply connected orthogonal polygon in the plane, and let n be fixed, n ≥ 1. If every two points of S are visible via staircase n-paths from a common point of S, then S is starshaped via staircase (n + 1)-paths. Moreover, the associated staircase (n + 1)-kernel is staircase (n + 1)-convex. The number two is best possible, and the number n + 1 is best possible for n ≥ 2. 相似文献
17.
Marilyn Breen 《Aequationes Mathematicae》2009,78(3):297-308
Let C be an orthogonal polygon in the plane, bounded by a simple closed curve, and assume that C is starshaped via staircase paths. Let
P í \mathbbR2\(int C)P \subseteq {\mathbb{R}}^2\backslash({\rm int} C). If every four points of P see a common boundary point of C via staircase paths in
\mathbbR2\(int C){\mathbb{R}}^2\backslash({\rm int} C), then there is a boundary point b of C such that every point of P sees b (via staircase paths in
\mathbbR2\(int C){\mathbb{R}}^2\backslash({\rm int} C)). The number four is best possible, even if C is orthogonally convex. 相似文献
18.
Marilyn Breen 《Monatshefte für Mathematik》2000,130(1):1-5
Let , where is an open connected subset of some linear topological space, such that S contains all triangular regions whose (relative) boundaries lie in S. If some finite subset T of S has locally maximal visibility in S, then . Hence S is a finite union of starshaped sets whose kernels are determined by T. An analogous result holds for S open. Moreover, counterexamples show that neither the requirement on triangular regions nor the restriction to a finite set
T can be deleted.
(Received 7 September 1998; in revised form 25 October 1999) 相似文献
19.
We establish the following Helly-type theorem: Let ${\cal K}$ be a family of
compact sets in $\mathbb{R}^d$. If every d + 1 (not necessarily
distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel
contains a translate of set A, then
$\cap \{ K : K\; \hbox{in}\; {\cal K} \}$ also is a starshaped set whose kernel contains a
translate of A. An analogous result holds
when ${\cal K}$ is a finite family of closed sets in $\mathbb{R}^d$.
Moreover, we have the following planar result: Define function f on
$\{0, 1, 2\}$ by f(0) = f(2) = 3, f(1) = 4. Let ${\cal K}$ be a finite
family of closed sets in the plane. For k = 0, 1, 2, if every f(k)
(not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set
whose kernel has dimension at least k,
then $\cap \{K : K\; \hbox{in}\; {\cal K}\}$ also is a starshaped set whose kernel has
dimension at least k. The number f(k) is best
in each case.Received: 4 June 2002 相似文献
20.
Marilyn Breen 《Journal of Geometry》1983,21(1):42-52
Let S be a compact, connected, locally starshaped set in Rd, S not convex. For every point of local nonconvexity q of S, define Aq to be the subset of S from which q is clearly visible via S. Then ker S = {conv Aq: q lnc S}. Furthermore, if every d+1 points of local nonconvexity of S are clearly visible from a common d-dimensional subset of S, then dim ker S = d. 相似文献