共查询到20条相似文献,搜索用时 15 毫秒
1.
Chuanming Zong 《数学学报(英文版)》2016,32(1):124-136
In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied. 相似文献
2.
R. Komendarczyk 《Transactions of the American Mathematical Society》2006,358(6):2399-2413
In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a two-dimensional consequence of this fact: there is a close relationship between the topology of the contact structure on a convex surface in the 3-manifold (the dividing curves) and the nodal curves of Laplacian eigenfunctions on that surface. Motivated by this relationship, we consider a topological version of Payne's conjecture for the free membrane problem. We construct counterexamples to Payne's conjecture for closed Riemannian surfaces. In light of the correspondence between the nodal lines and dividing curves, we interpret the conjecture in terms of the tight versus overtwisted dichotomy for contact structures.
3.
《Mathematische Nachrichten》2018,291(10):1502-1513
We obtain sharp estimates for a generalized Zalcman coefficient functional with a complex parameter for the Hurwitz class and the Noshiro–Warschawski class of univalent functions as well as for the closed convex hulls of the convex and starlike functions by using an inequality from [6]. In particular, we generalize an inequality proved by Ma for starlike functions and answer a question from his paper [17]. Finally, we prove an asymptotic version of the generalized Zalcman conjecture for univalent functions and discuss various related or equivalent statements which may shed further light on the problem. 相似文献
4.
V. A. Zalgaller 《Siberian Mathematical Journal》2009,50(5):846-855
We select a class of pyramids of a particular shape and propose a conjecture that precisely these pyramids are of greatest
surface area among the closed convex polyhedra having evenly many vertices and the unit geodesic diameter. We describe the
geometry of these pyramids. The confirmation of our conjecture will solve the “doubly covered disk” problem of Alexandrov.
Through a connection with Reuleaux polygons we prove that on the plane the convex n-gon of unit diameter, for odd n, has greatest area when it is regular, whereas this is not so for even n. 相似文献
5.
Let K⊂Rn be a convex body (a compact, convex subset with non-empty interior), ΠK its projection body. Finding the least upper bound, as K ranges over the class of origin-symmetric convex bodies, of the affine-invariant ratio V(ΠK)/V(K)n−1, being called Schneider's projection problem, is a well-known open problem in the convex geometry. To study this problem, Lutwak, Yang and Zhang recently introduced a new affine invariant functional for convex polytopes in Rn. For origin-symmetric convex polytopes, they posed a conjecture for the new functional U(P). In this paper, we give an affirmative answer to the conjecture in Rn, thereby, obtain a modified version of Schneider's projection problem. 相似文献
6.
We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman's reverse Brunn–Minkowski inequality. 相似文献
7.
Beata Laszkiewicz 《Linear algebra and its applications》2008,428(4):1031-1040
We investigate the properties of the approximation of a matrix by matrices whose spectra are in a closed convex set of the complex plane. We explain why the Khalil and Maher characterization of an approximant, which spectrum is in a strip, is not quite correct. We prove that their characterization is valid but for another kind of approximation. We formulate a conjecture which leads to some algorithm for computing approximants. The conjecture is motivated by numerical experiments and some theoretical considerations. Separately we consider the approximation of normal matrices. 相似文献
8.
Error bounds for set inclusions 总被引:1,自引:0,他引:1
郑喜印 《中国科学A辑(英文版)》2003,46(6):750-763
A variant of Robinson-Ursescu Theorem is given in normed spaces. Several error bound theorems for convex inclusions are proved and in particular a positive answer to Li and Singer's conjecture is given under weaker assumption than the assumption required in their conjecture. Perturbation error bounds are also studied. As applications, we study error bounds for convex inequality systems. 相似文献
9.
Lutwak, Yang and Zhang established the Orlicz centroid inequality for convex bodies and conjectured that their inequality can be extended to star bodies. In this paper, we confirm this conjecture. 相似文献
10.
In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered. 相似文献
11.
Yoshio Okamoto 《Discrete Mathematics》2008,308(17):3836-3846
A generalized convex shelling was introduced by Kashiwabara et al. for their representation theorem of convex geometries. Motivated by the work by Edelman and Reiner, we study local topology of the free complex of a two-dimensional separable generalized convex shelling. As a result, we prove a deletion of an element from such a complex is homotopy equivalent to a single point or two distinct points, depending on the dependency of the element to be deleted. Our result resolves an open problem by Edelman and Reiner for this case, and it can be seen as a first step toward the complete resolution from the viewpoint of a representation theorem for convex geometries by Kashiwabara et al. 相似文献
12.
A convex geometry is a closure system whose closure operator satisfies the anti-exchange property. As is described in Sagan’s
survey paper, characteristic polynomials factorize over nonnegative integers in several situations. We show that the characteristic
polynomial of a 2-tight convex geometry K factorizes over nonnegative integers if the clique complex of the nbc-graph of K is pure and strongly connected. This factorization theorem is new in the sense that it does not belong to any of the three
categories mentioned in Sagan’s survey.
Received September 25, 2005 相似文献
13.
Ge Xiong 《Advances in Mathematics》2010,225(6):3214-3228
Lutwak, Yang and Zhang defined the cone volume functional U over convex polytopes in Rn containing the origin in their interiors, and conjectured that the greatest lower bound on the ratio of this centro-affine invariant U to volume V is attained by parallelotopes. In this paper, we give affirmative answers to the conjecture in R2 and R3. Some new sharp inequalities characterizing parallelotopes in Rn are established. Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes in Rn is provided. 相似文献
14.
15.
It was a long-standing conjecture in finite geometry that a Desarguesian plane of odd order contains no maximal arcs. A rather inaccessible and long proof was given recently by the authors in collaboration with Mazzocca. In this paper a new observation leads to a greatly simplified proof of the conjecture.
16.
Ben Andrews 《Inventiones Mathematicae》1999,138(1):151-161
We prove Firey’s 1974 conjecture that convex surfaces moving by their Gauss curvature become spherical as they contract to
points.
Oblatum 20-VII-1998 & 19-III-1999 / Published online: 6 July 1999 相似文献
17.
Karl Abrahamson 《Discrete and Computational Geometry》1990,5(1):409-419
Under two definitions of random convex polygons, the expected modality of a random convex polygon grows without bound as the number of vertices grows. This refutes a conjecture of Aggarwal and Melville. 相似文献
18.
Gennadi Kasparov 《Advances in Mathematics》2006,206(1):1-56
The coarse geometric Novikov conjecture provides an algorithm to determine when the higher index of an elliptic operator on a noncompact space is nonzero. The purpose of this paper is to prove the coarse geometric Novikov conjecture for spaces which admit a (coarse) uniform embedding into a uniformly convex Banach space. 相似文献
19.
N. Yu. Erokhovets 《Proceedings of the Steklov Institute of Mathematics》2009,266(1):120-132
Convex polytopes have interested mathematicians since very ancient times. At present, they occupy a central place in convex geometry, combinatorics, and toric topology and demonstrate the harmony and beauty of mathematics. This paper considers the problem of describing the f-vectors of simple flag polytopes, that is, simple polytopes in which any set of pairwise intersecting facets has nonempty intersection. We show that for each nestohedron corresponding to a connected building set, the h-polynomial is a descent-generating function for some class of permutations; we also prove Gal’s conjecture on the nonnegativity of γ-vectors of flag polytopes for nestohedra constructed over complete bipartite graphs. 相似文献
20.
Anm-transversal to a family of convex sets in the plane is anm-point set which intersects every members of the family. One of Grübaum’s conjectures says that a planar family of translates of a convex compact set has a 3-transversal provided that any two of its members intersect. Recently the conjecture has been proved affirmatively (see [4]). In the present paper we provide a different and straightforward proof for the conjecture for the family of translates of a closed trapezoid in the plane and give several concrete 3-transversals. 相似文献