共查询到20条相似文献,搜索用时 31 毫秒
1.
R. H. Aramyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2018,53(6):363-368
The problem of the sine representation for the support function of centrally symmetric convex bodies is studied. We describe a subclass of centrally symmetric convex bodies which is dense in the class of centrally symmetric convex bodies. Also, we obtain an inversion formula for the sine-transform. 相似文献
2.
Zsolt Lángi 《Israel Journal of Mathematics》2016,212(1):203-217
A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes–Thompson, and Gromov’s mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies. 相似文献
3.
In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space
\Bbb Rd{\Bbb R}^d
containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the
only extremal bodies, which is proved for all three dimensional and several prominent lattices. 相似文献
4.
Christina Bauer 《Geometriae Dedicata》1996,62(2):179-192
We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class. 相似文献
5.
In this paper we consider a new class of convex bodies which was introduced in [11]. This is the class of belt bodies, and it is a natural generalization of the class of zonoids (see the surveys [18, 28, 24]). While the class of zonoids is not dense in the family of all centrally symmetric, convex bodies, the class of belt bodies is dense in the set of all convex bodies. Nevertheless, we shall extend solutions of combinatorial problems for zonoids (cf. [2, 12]) to the class of belt bodies. Therefore, we first introduce the set of belt bodies by using zonoids as starting point. (To make the paper self-contained, a few parts of the approach from [11] are given repeatedly.) Second, complete solutions of three well-known (and generally unsolved) problems from the combinatorial geometry of convex bodies are given for the class of belt bodies. The first of these, connected with the names of I. Gohberg and H. Hadwiger, is the problem of covering a convex body with smaller homothetic copies, or the equivalent illumination problem. The second is the Szökefalvi-Nagy problem, which asks for the determination of the convex bodies whose families of translates have a given Helly dimension. The third problem concerns special fixing systems, a notion which is due to L. Fejes Tóth. These solutions consist of improved and more general approaches to recently solved problems (as in the case of the Helly-dimensional classification of belt bodies) or new results (as those concerning minimal fixing systems, providing also an answer to a problem of B. Grünbaum which is not only restricted to belt bodies). 相似文献
6.
We consider finite lattice ball packings with respect to parametric density and show that densest packings are attained in critical lattices if the number of translates and the density parameter are sufficiently large. A corresponding result is not valid for general centrally symmetric convex bodies. 相似文献
7.
Károly J. Böröczky 《Advances in Mathematics》2010,225(4):1914-1928
A stability version of the Blaschke-Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n?3 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established. 相似文献
8.
We consider finite lattice ball packings with respect to parametric density and show that densest packings are attained in critical lattices if the number of translates and the density parameter are sufficiently large. A corresponding result is not valid for general centrally symmetric convex bodies.The second author was partially supported by a DAAD Postdoc fellowship and the hospitality of Peking University during his work. 相似文献
9.
A covering of the Euclidean plane by a polygon P is a system of translated copies of P whose union is the plane, and a packing of P in the plane is a system of translated copies of P whose interiors are disjoint. A lattice covering is a covering in which the translates are defined by the points of a lattice, and a lattice packing is defined similarly. We show that, given a convex polygon P with n vertices, the densest lattice packing of P in the plane can be found in O(n) time. We also show that the sparsest lattice covering of the plane by a centrally symmetric convex polygon can be solved in O(n) time. Our approach utilizes results from classical geometry that reduce these packing and covering problems to the problems of finding certain extremal enclosed figures within the polygon. 相似文献
10.
Periodica Mathematica Hungarica - In his paper Á. G. Horváth posed two isoperimetric type questions for extremal polyhedra with respect to a given lattice L. He solved the problems in the... 相似文献
11.
We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, those of sections of convex bodies and of sections of their circumscribed cylinders. For L d a convex body, we take n random segments in L and consider their 'Minkowski average' D. For fixed n, the pth moments of V(D) (1 p < ) are minimized, for V (L) fixed, by the ellipsoids. For k = 2 and fixed n, the pth moment of V(D) is maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies. 相似文献
12.
Vicent Caselles 《Journal of Functional Analysis》2010,259(6):1491-1516
The aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context. 相似文献
13.
In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space
containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the
only extremal bodies, which is proved for all three dimensional and several prominent lattices.
Authors’ addresses: M. A. Hernández Cifre, Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia,
Spain; A. Schürmann, Institut für Algebra und Geometrie, Otto-von-Guericke Universit?t Magdeburg, 39106 Magdeburg, Germany;
F. Vallentin, Centrum voor Wiskunde en Informatica (CWI), Kruislaan 413, 1098 SJ Amsterdam, The Netherlands 相似文献
14.
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
15.
该文先介绍一些中国数学家在几何不等式方面的工作.作者用积分几何中著名的Poincarè公式及Blaschke公式估计一随机凸域包含另一域的包含测度, 得到了经典的等周不等式和Bonnesen -型不等式.还得到了一些诸如对称混合等周不等式、Minkowski -型和Bonnesen -型对称混合等似不等式在内的一些新的几何不等式.最后还研究了Gage -型等周不等式以及Ros -型等周不等式. 相似文献
16.
We establish a close relationship between isoperimetric inequalities for convex bodies and asymptotic shapes of large random
polytopes, which arise as cells in certain random mosaics in d-dimensional Euclidean space. These mosaics are generated by Poisson hyperplane processes satisfying a few natural assumptions
(not necessarily stationarity or isotropy). The size of large cells is measured by a class of general functionals. The main
result implies that the asymptotic shapes of large cells are completely determined by the extremal bodies of an inequality
of isoperimetric type, which connects the size functional and the expected number of hyperplanes of the generating process
hitting a given convex body. We obtain exponential estimates for the conditional probability of large deviations of zero cells
from asymptotic or limit shapes, under the condition that the cells have large size.
This work was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.
Received: May 2005 Accepted: September 2005 相似文献
17.
H. Groemer 《Discrete and Computational Geometry》1990,5(1):357-364
It is well known that ann-dimensional convex body permits a lattice packing of density 1 only if it is a centrally symmetric polytope of at most 2(2
n
–1) facets. This article concerns itself with the associated stability problem whether a convex body that permits a packing of high density is in some sense close to such a polytope. Several inequalities that address this stability problem are proved. A corresponding theorem for coverings by two-dimensional convex bodies is also proved.Supported by National Science Foundation Research Grants DMS 8300825 and DMS 8701893. 相似文献
18.
Paul R. Goodey 《Geometriae Dedicata》1986,20(3):295-305
This work is motivated by the stability problem for projection bodies of centrally symmetric convex bodies. The problem asks whether the closeness of two projection bodies implies closeness of the original bodies. Here we will give a negative answer and, in fact, work in a setting rather more general than that of the original problem. 相似文献
19.
For a centrally symmetric convex
and a covering lattice L for K, a lattice polygon P is called a covering polygon, if
. We prove that P is a covering polygon, if and only if its boundary bd(P) is covered by (L ∩ P) + K. Further we show that this characterization is false for non-symmetric planar convex bodies and
in Euclidean d–space, d ≥ 3, even for the unit ball K = B
d.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
20.
The title problem is examined for plane strain deformationsof an ideal fibre-reinforced body which has a convex cross-sectionwith an axis of symmetry either parallel or normal to the fibreaxis. The solution to the problem is found to be easily obtainedeven for symmetric bodies of a general shape. This is in markedcontrast with the corresponding problems for an anisotropicelastic body and enables approximate solutions for stronglyanisotropic bodies to be derived without difficulty. 相似文献