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1.
In this paper, we first establish a constant rank theorem for the second fundamental form of the convex level sets of harmonic functions in space forms. Applying the deformation process, we prove that the level sets of the harmonic functions on convex rings in space forms are strictly convex. Moreover, we give a lower bound for the Gaussian curvature of the convex level sets of harmonic functions in terms of the Gaussian curvature of the boundary and the norm of the gradient on the boundary. 相似文献
2.
In this paper, we present a new class of alternative theorems for SOS-convex inequality systems without any qualifications. This class of theorems provides an alternative equations in terms of sums of squares to the solvability of the given inequality system. A strong separation theorem for convex sets, described by convex polynomial inequalities, plays a key role in establishing the class of alternative theorems. Consequently, we show that the optimal values of various classes of robust convex optimization problems are equal to the optimal values of related semidefinite programming problems (SDPs) and so, the value of the robust problem can be found by solving a single SDP. The class of problems includes programs with SOS-convex polynomials under data uncertainty in the objective function such as uncertain quadratically constrained quadratic programs. The SOS-convexity is a computationally tractable relaxation of convexity for a real polynomial. We also provide an application of our theorem of the alternative to a multi-objective convex optimization under data uncertainty. 相似文献
3.
In this article, we deal with some computational aspects of geodesic convex sets. Motzkin-type theorem, Radon-type theorem, and Helly-type theorem for geodesic convex sets are shown. In particular, given a finite collection of geodesic convex sets in a simple polygon and an “oracle,” which accepts as input three sets of the collection and which gives as its output an intersection point or reports its nonexistence; we present an algorithm for finding an intersection point of this collection. 相似文献
4.
G. E. Ivanov 《Mathematical Notes》2006,79(1-2):55-78
In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game. 相似文献
5.
Jeroen Kuipers Dries Vermeulen Mark Voorneveld 《International Journal of Game Theory》2010,39(4):585-602
The Shapley–Ichiishi result states that a game is convex if and only if the convex hull of marginal vectors equals the core.
In this paper, we generalize this result by distinguishing equivalence classes of balanced games that share the same core
structure. We then associate a system of linear inequalities with each equivalence class, and we show that the system defines
the class. Application of this general theorem to the class of convex games yields an alternative proof of the Shapley–Ichiishi
result. Other applications range from computation of stable sets in non-cooperative game theory to determination of classes
of TU games on which the core correspondence is additive (even linear). For the case of convex games we prove that the theorem
provides the minimal defining system of linear inequalities. An example shows that this is not necessarily true for other
equivalence classes of balanced games. 相似文献
6.
B. Klartag 《Journal of Functional Analysis》2007,245(1):284-310
We investigate the rate of convergence in the central limit theorem for convex sets established in [B. Klartag, A central limit theorem for convex sets, Invent. Math., in press. [8]]. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets. 相似文献
7.
V. Klee extended a well-known theorem of Minkowski to non-compact convex sets. We generalize Minkowski’s theorem to convex sets which are not necessarily closed. 相似文献
8.
We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests
a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turns out to
be an oriented matroid. We also give new upper bounds on the Erdős–Szekeres theorem in the context of convex bodies. 相似文献
9.
Kam-Chau Wong 《Mathematical Logic Quarterly》1996,42(1):564-568
We prove in recursive analysis an existence theorem for computable minimizers of convex computable continuous real-valued functions, and a computable separation theorem for convex sets in ?m. Mathematics Subject Classification: 03F60, 52A40. 相似文献
10.
以Banach空间的一般凸集为研究对象,将Banach空间的凸性研究推广到了内部非空的凸集上.打破了从单位球出发研究Banach空间几何的具有局限性的研究方法,给出了严格凸集的若干特征刻画及性质,并得到了严格凸集和光滑集之间的对偶定理. 相似文献
11.
《Computational Geometry》2005,30(2):129-144
A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of “convexity” shared by some objects including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, and show that every convex geometry can be represented as a generalized convex shelling. This is “the representation theorem for convex geometries” analogous to “the representation theorem for oriented matroids” by Folkman and Lawrence. An important feature is that our representation theorem is affine-geometric while that for oriented matroids is topological. Thus our representation theorem indicates the intrinsic simplicity of convex geometries, and opens a new research direction in the theory of convex geometries. 相似文献
12.
We prove that a collection of compact convex sets of bounded diameters in
that is unbounded in k independent directions has a k-flat transversal for k<d if and only if every d+1 of the sets have a k-transversal. This result generalizes a theorem of Hadwiger(–Danzer–Grünbaum–Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d−1. 相似文献
13.
László Losonczi 《Journal of Mathematical Analysis and Applications》1973,44(3):701-709
In this study we present an important theorem of the alternative involving convex functions and convex cones. From this theorem we develop saddle value optimality criteria and stationary optimality criteria for convex programs. Under suitable constraint qualification we obtain a generalized form of the Kuhn-Tucker conditions. We also use the theorem of the alternative in developing an important duality theorem. No duality gaps are encountered under the constraint qualification imposed earlier and the dual problem always possesses a solution. Moreover, it is shown that all constraint qualifications assure that the primal problem is stable in the sense used by Gale and others. The notion of stability is closely tied up with the positivity of the lagrangian multiplier of the objective function. 相似文献
14.
我们根据一般化凸空间上的KKM型定理得到了截口定理,然后作为它的应用讨论了若干个择一不等式.最后,引进了一个具体的一般化凸空间并在该空间上讨论了择一不等式解的存在性问题. 相似文献
15.
A STRONG OPTIMIZATION THEOREM IN LOCALLY CONVEX SPACES 总被引:1,自引:0,他引:1
This paper presents a geometric characterization of convex sets in locally convex spaces on which a strong optimization theorem of the Stegall-type holds, and gives Collier's theorem of w Asplund spaces a localized setting. 相似文献
16.
J Vangeldère 《Journal of Mathematical Analysis and Applications》1977,60(1):36-46
First we give some elementary properties of the core of a subset relative to a linear subspace. Then we prove a theorem on the frank separation of two convex sets. This theorem admits as particular cases the known theorems on the frank separation and introduces new cases. Finally, we provide a very general version of the Hahn-Banach theorem in an analytic form. 相似文献
17.
Based on a study of a minimization problem, we present the following results applicable to possibly nonconvex sets in a Banach space: an approximate projection result, an extended extremal principle, a nonconvex separation theorem, a generalized Bishop-Phelps theorem and a separable point result. The classical result of Dieudonné (on separation of two convex sets in a finite-dimensional space) is also extended to a nonconvex setting. 相似文献
18.
J. Ch. Pomerol 《Mathematical Programming》1980,19(1):352-355
We give a new minisup theorem for noncompact strategy sets. Our result is of the type of the Matthies-Strang-Christiansen minimax theorem where the hyperplane should be replaced by any closed convex set. As an application, we derive a slight generalization of the Matthies-Strang-Christiansen minimax theorem. 相似文献
19.
模糊集的表现定理是模糊数学的最基本理论.在表现定理的基础上,对各种模糊量:包括凸模糊量、正规模糊量、正规凸模糊量、凸有界模糊量、模糊数、有限模糊数、对称模糊数的表现定理进行了深入的研究,从而建立了不同类型模糊量与普通集合之间的联系. 相似文献