共查询到20条相似文献,搜索用时 15 毫秒
1.
Marilyn Breen 《Journal of Geometry》1988,32(1-2):1-12
Let S be a subset of the plane. In case (int cl S) S = , then S is finitely starlike if and only if every 4 points of S see via S a common point. In case (int cl S) S has at most countably many components, each a singleton set, then S is finitely starlike if and only if every 5 points of S see via S a common point. Each of the numbers 4 and 5 is best possible. Examples show that these results fail without suitable restrictions on (int cl S) S. Moreover, a final example shows that if a general Krasnosel'skii number . exists to characterize finitely starlike sets in the plane, then > 9. 相似文献
2.
Stefano Campi 《Geometriae Dedicata》1992,43(1):71-81
The paper deals with the following question: Among the convex plane sets of fixed isoperimetric deficit, which are the sets of maximum translative deviation from the circular shape? The answer is given for the cases in which the deviation is measured either by the translative Hausdorff metric or by the translative symmetric difference metric. 相似文献
3.
We introduce a new combinatorial object, the double-permutation sequence, and use it to encode a family of mutually disjoint
compact convex sets in the plane in a way that captures many of its combinatorial properties. We use this encoding to give
a new proof of the Edelsbrunner-Sharir theorem that a collection of n compact convex sets in the plane cannot be met by straight lines in more than 2n-2 combinatorially distinct ways. The encoding generalizes the authors’ encoding of point configurations by “allowable sequences”
of permutations. Since it applies as well to a collection of compact connected sets with a specified pseudoline arrangement
of separators and double tangents, the result extends the Edelsbrunner-Sharir theorem to the case of geometric permutations
induced by pseudoline transversals compatible with .
Supported in part by NSA grant MDA904-03-I-0087 and PSC-CUNY grant 65440-0034.
Supported in part by NSF grant CCR-9732101. 相似文献
4.
We prove that for a measurable subset of S
n–1 with fixed Haar measure, the volume of its convex hull is minimized for a cap (i.e. a ball with respect to the geodesic measure). We solve a similar problem for symmetric sets and n=2, 3. As a consequence, we deduce a result concerning Gaussian measures of dilatations of convex, symmetric sets in R
2 and R
3.Partially supported by KBN (Poland), Grant No. 2 1094 91 01. 相似文献
5.
Branko ?urgus 《Discrete Applied Mathematics》2007,155(13):1774-1792
Let S be a finite set with m elements in a real linear space and let JS be a set of m intervals in R. We introduce a convex operator co(S,JS) which generalizes the familiar concepts of the convex hull, , and the affine hull, , of S. We prove that each homothet of that is contained in can be obtained using this operator. A variety of convex subsets of with interesting combinatorial properties can also be obtained. For example, this operator can assign a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For two types of families JS we give two different upper bounds for the number of vertices of the polytopes produced as co(S,JS). Our motivation comes from a recent improvement of the well-known Gauss-Lucas theorem. It turns out that a particular convex set co(S,JS) plays a central role in this improvement. 相似文献
6.
Let C be a convex body in the Euclidean plane. The relative distance of points p and q is twice the Euclidean distance of p and q divided by the Euclidean length of a longest chord in C with the direction, say, from p to q. We prove that, among any seven points of a plane convex body, there are two points at relative distance at most one, and
one cannot be replaced by a smaller value. We apply our result to determine the diameter of point sets in normed planes.
Zsolt Lángi: Partially supported by the Hung. Nat. Sci. Found. (OTKA), grant no. T043556 and T037752 and by the Alberta Ingenuity
Fund. 相似文献
7.
H. Groemer 《Aequationes Mathematicae》1981,22(1):215-222
In the euclidean planeE
2 letS
1,S
2, ... be a sequence of strips of widthsw
1,w
2, .... It is shown thatE
2 can be covered by translates of the stripsS
i if w
1
3/2
= . Further results concern conditions in order that a compact convex domain inE
2 can be covered by translates ofS
1,S
2, ....This research was supported by National Science Foundation Research Grant MCS 76-06111. 相似文献
8.
9.
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
10.
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
11.
Marilyn Breen 《Archiv der Mathematik》2005,84(3):282-288
Let k and d be fixed integers, 0kd, and let
be a collection of sets in
If every countable subfamily of
has a starshaped intersection, then
is (nonempty and) starshaped as well. Moreover, if every countable subfamily of
has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of
is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004 相似文献
12.
Kurt Leichtweiß 《Journal of Geometry》2003,78(1-2):92-121
Analogue to the definition $K + L := \bigcup_{x\in K}(x + L)$ of the
Minkowski addition in the euclidean geometry it is proposed to define the
(noncommutative) addition $K \vdash L := \bigcup_{0\, \leqsl\, \rho\,\leqsl\,
a(\varphi),0\,\leqsl\,\varphi\,<\, 2\pi}T_{\rho}^{(\varphi)}(L)$ for compact,
convex and smoothly bounded sets K and
L in the hyperbolic plane $\Omega$
(Kleins model). Here $\rho = a(\varphi)$ is the representation of the boundary
$\partial$ K in geodesic polar coordinates
and $T_{\rho}^{(\varphi)}$ is the hyperbolic translation of $\Omega$ of length
$\rho$ along the line through the origin o of
direction $\varphi$. In general this addition does not preserve
convexity but nevertheless we may prove as main results: (1) $o \in$ int
$K, o \in$ int L and K,L horocyclic convex imply the strict
convexity of $K \vdash L$, and (2) in this case there exists a hyperbolic mixed
volume $V_h(K,L)$ of K and L which has a representation by a suitable
integral over the unit circle. 相似文献
13.
14.
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
15.
16.
On pairs of vectors achieving the maximal angle of a convex cone 总被引:1,自引:1,他引:0
In this paper we explore the concept of antipodality relative to a closed convex cone . The problem under consideration is that of finding a pair of unit vectors in K achieving the maximal angle of the cone. We mention also a few words on the attainability of critical angles. By way of application
of the general theory, we briefly discuss the problem of estimating the radius of pointedness of a cone. 相似文献
17.
《Quaestiones Mathematicae》2013,36(2):271-283
AbstractMotivated by the notion of volume difference functions, we introduce quotient functions of dual quermassintegrals and establish Brunn-Minkowski type inequalities for them, which have several recent results as special cases. 相似文献
18.
Marilyn Breen 《Aequationes Mathematicae》2004,67(3):263-275
Summary.
We establish the following Helly-type result for infinite families
of starshaped sets in
Define the function f on
{1, 2} by
f(1) = 4,
f(2) = 3.
Let
be a fixed positive number, and let
be a uniformly bounded family of compact sets
in the plane. For k = 1, 2, if every
f(k)
(not necessarily distinct) members of
intersect in a starshaped set whose
kernel contains a k-dimensional
neighborhood of radius
, then
is a starshaped set whose kernel is at least
k-dimensional.
The number f(k) is best in each case.
In addition, we present a few results concerning the dimension of
the kernel in an intersection of starshaped sets in
Some of these involve finite families of sets, while others
involve infinite families and make use of the Hausdorff metric. 相似文献
19.
Marilyn Breen 《Journal of Geometry》1987,28(1):80-85
Let S be a compact set in Rd. Let p be a fixed point of S and let k be a fixed integer, 1 k <d. Then S is starshaped with p ker S if and only if for every k-dimensional flat F through p, F S is starshaped. Moreover, an analogue of this result holds for unions of starshaped sets as well. 相似文献
20.
Marilyn Breen 《Geometriae Dedicata》1992,42(2):215-222
Let S be a compact set in the plane. If every three points of S are illuminated clearly by some translate of the compact convex set T, then there is a translate of T which illumines every point of S. Various analogues hold for translates of flats in R
das well.Supported in part by NSF grant DMS-8705336. 相似文献