共查询到20条相似文献,搜索用时 0 毫秒
1.
Daniel A. Klain 《Advances in Mathematics》2004,185(2):270-288
The traditional solution to the Minkowski problem for polytopes involves two steps. First, the existence of a polytope satisfying given boundary data is demonstrated. In the second step, the uniqueness of that polytope (up to translation) is then shown to follow from the equality conditions of Minkowski's inequality, a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context. In this article we adapt the classical argument to prove both the existence theorem of Minkowski and his mixed volume inequality simultaneously, thereby providing a new proof of Minkowski's inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry. 相似文献
2.
The logarithmic Minkowski problem asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the cone-volume measure of a convex body. This problem was solved recently by Böröczky, Lutwak, Yang and Zhang for even measures (Böröczky et al. (2013) [8]). This paper solves the case of discrete measures whose supports are in general position. 相似文献
3.
Michael J. Todd 《Journal of Combinatorial Theory, Series B》1976,20(3):229-242
Primoids and duoids are collections of subsets of a fixed finite set with a natural generalization of a pivoting property of convex polytopes. This structure is precisely what is necessary for the application of complementary pivoting algorithms. This paper investigates the combinatorial structure of primoids and duoids, showing them to form the circuits and cocircuits of a binary matroid. This matroid is then compared with the simplicial geometries of Crapo and Rota. 相似文献
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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.
A. Schatteman 《Geometriae Dedicata》1990,34(3):229-242
A vertex v of a convex polygon P is called minimal (respectively maximal) if the circle going through v and its neighbouring vertices encloses the interior of P (respectively has no vertex of P in its interior) The main result of this paper is a generalization to the convex polytopes of R
d
of the following theorem: Every convex polygon has at least two minimal and two maximal vertices The proof uses a duality theory which translates some spherical properties of a convex polytope of R
d
into combinatorial properties of a convex polyhedron of R
d+1. 相似文献
7.
William Jockusch 《Discrete and Computational Geometry》1993,9(1):159-163
We construct a family of cubical polytypes which shows that the upper bound on the number of facets of a cubical polytope (given a fixed number of vertices) is higher than previously suspected. We also formulate a lower bound conjecture for cubical polytopes.This paper was researched and written while the author was a graduate student at MIT. The author was partially supported by an NSF Graduate Fellowship. 相似文献
8.
The purpose of this paper is to prove that the Mirković–Vilonen (MV) polytope corresponding to the tensor product of two arbitrary
MV polytopes is contained, as a set, in the Minkowski sum of these two MV polytopes. This result generalizes the one in our
previous paper, which was obtained under the assumption that the first tensor factor is an extremal MV polytope. 相似文献
9.
We present 35 open problems on combinatorial, geometric and algebraic aspects of -orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work but has not been formalized before. 相似文献
10.
Summary In this paper, we present variants of a convergent projection and contraction algorithm [25] for solving projection problems over polytope. By using the special struture of the projection problems, an iterative algorithm with constant step-size is given, which is globally linearly convergent. These algorithms are simple to implement and each step of the method requires only a few matrix-vector multiplications. Especially, for minimums norm problems over transportation or general network polytopes onlyO(n) additions andO(n) multiplications are needed at each iteration. Numerical results for randomly generated test problems over network polytopes, up to 10000 variables, indicate that the presented algorithms are simple and efficient even for large problems. 相似文献
11.
Geometriae Dedicata - 相似文献
12.
We consider the problem of updating input-output matrices, i.e., for given (m,n) matrices A ? 0, W ? 0 and vectors u ? m, v?n, find an (m,n) matrix X ? 0 with prescribed row sums Σnj=1Xij = ui (i = 1,…,m) and prescribed column sums Σmi=1Xij = vj (j = 1,…,n) which fits the relations Xij = Aij + λiWij + Wij + Wijμj for all i,j and some λ?m, μ?n. Here we consider the question of existence of a solution to this problem, i.e., we shall characterize those matrices A, W and vectors u,v which lead to a solvable problem. Furthermore we outline some computational results using an algorithm of [2]. 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.
V. I. Oliker 《Mathematical Notes》1973,13(1):26-30
We prove uniqueness in the Minkowski problem for open surfaces with spherical image greater than a hemisphere. We extend the class of regions determined earlier by Volkov and the author in which the Christoffel problem has a unique solution.Translated from Matematicheskie Zametki, Vol. 13, No. 1, pp. 41–49, January, 1973.In conclusion the author wishes to express his gratitude to Yu. D. Burago, Yu. A. Volkov, and V. G. Maz'e for valuable advice. 相似文献
15.
S. S. Kutateladze 《Journal of Applied and Industrial Mathematics》2013,7(2):209-214
Under study are some vector optimization problems over the space of Minkowski balls, i.e., symmetric convex compact subsets in Euclidean space. A typical problem requires to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a symmetric convex body $\mathfrak{x}$ , we try to maximize the volume of $\mathfrak{x}$ and minimize the width of $\mathfrak{x}$ simultaneously. 相似文献
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《Discrete Mathematics》2006,306(10-11):886-904
19.
A convexd-polytype inE
d is called semiregular, if its facets are regular and its vertices equivalent. A list of semiregular polytopes ford≥4 is known since 1900. Recently it has been proved by П. В. Макаров [сб. Вопр. дискр. геом., Мат. исслед. ИМ АН Молд. ССР
вып. 103, C. 139–150, Кищинев 1988], that this list is complete ford=4. We present here a simple proof for that this list is complete in any dimension. 相似文献
20.
《Operations Research Letters》1987,6(5):239-241
A subgraph F of graph G is called a perfectly matchable subgraph if F contains a set of independent edges convering all the vertices in F. The convex hull of the incidence vectors of perfectly matchable subgraphs of G is a 0–1 polytope. We characterize the adjacency of vertices on such polytopes. We also show that when G is bipartite, the separation problem for such polytones can be solved by maximum flow algorithms. 相似文献