Convex equipartitions via Equivariant Obstruction Theory

We describe a regular cell complex model for the configuration space F(? ^d , n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.     

On the equipartitions of finite groups

Summary It is proved that the p- groups of submaximal class do not admit equipartitions other than the atomic (Theorem 3.2). This result is then wed to give a complete classification of the groups of order p⁶ admitting nonatomic equipartitions (Theorem 5.1).     

Convex billiards and a theorem by E. Hopf

Supported by the Colton Fellowship     

Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l.     

Convex Defining Functions for Convex Domains

We give three proofs of the fact that a smoothly bounded, convex domain in ℝ ⁿ has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.     

A stable theorem of the alternative: an extension of the gordan theorem

A theorem with a number of equivalent alternatives is proposed as an extension of the classical Gordan theorem of the alternative. The theorem can handle nonzero unrestricted variables which cannot be directly treated by ordinary theorems of the alternative. Like the Gordan theorem, the extended theorem has the stability feature that small perturbations in the data will not invalidate an alternative that is in force. The theorem has useful applications in establishing the boundedness and uniqueness of feasible points of polyhedral sets and of solutions to linear programming problems.     

Convex Dimension of Locally Planar Convex Geometries

We prove that convex geometries of convex dimension n that satisfy two properties satisfied by nondegenerate sets of points in the plane, may have no more than 2 ^n-1 points. We give examples of such convex geometries that have n \choose 4 + n \choose 2 + n \choose 0 points. Received June 7, 1999, and in revised form April 18, 2000. Online publication September 22, 2000.     

Convex operators: some subdifferentiability results

The subdifferentiability properties of convex operators are intimately related to the properties of the ordered (topological) vector spaces they range in. This relation is investigated for various types of subdifferentiability notions     

Convex billiards

This article deals with (partly rather peculiar) properties of typical convex billiards. In particular, convex caustics, trajectories which terminate on the boundary and dense trajectories are investigated.Dedicated to Professor Curt Christian on the occasion of his 70th birthday     

Derivation of the Cartwright-Levinson theorem from the Kolmogorov theorem

One says that an entire function f of finite exponential type belongs to the Cartwright class C, if Let N₊(r)(N_–(r)) denote the number of zeros of the function f in the disk ¦z¦R, such that Re z0 (Re z<0, respectively). We give a simple derivation of the folfollowing result of importance in the theory of entire functions, from a weak type Kolmogorov inequality.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 76–86, 1984.I am happy to note that this paper was written in connection with the 50th birthday of my friend V. P. Khavin.     

Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions

We characterize the class of those closed convex sets which have a barrier cone with a nonempty interior. As a consequence, we describe the set of those proper extended-real-valued functionals for which the domain of their Fenchel conjugate has a nonempty interior. As an application, we study the stability of the solution set of a semi-coercive variational inequality.     

On Bayes' theorem and the inverse Bernoulli theorem

A recent assertion by S. M. Stigler that Thomas Bayes was perhaps anticipated in the discovery of the result that today bears his name is exposed to further scrutiny here. The distinction between Bayes' theorem and the inverse Bernoulli theorem is examined, and pertinent early writings on this matter are discussed. A careful examination of the difference between these two theorems leads to the conclusion that a result given by David Hartley in 1749 is more in line with the inverse Bernoulli theorem than with Bayes' result, and it is suggested that there is not sufficient evidence to remove Bayes from his place as originator of the method adopted.     

On the Maximization of (not necessarily) Convex Functions on Convex Sets

The global solutions of the problem of maximizing a convex function on a convex set were characterized by several authors using the Fenchel (approximate) subdifferential. When the objective function is quasiconvex it was considered the differentiable case or used the Clarke subdifferential. The aim of the present paper is to give necessary and sufficient optimality conditions using several subdifferentials adequate for quasiconvex functions. In this way we recover almost all the previous results related to such global maximization problems with simple proofs.     

Stability of the geometric Ekeland variational principle: Convex case

In geometric terms, the Ekeland variational principle says that a lower-bounded proper lower-semicontinuous functionf defined on a Banach spaceX has a point (x ₀,f(x ₀)) in its graph that is maximal in the epigraph off with respect to the cone order determined by the convex coneK _λ = {(x, α) ∈X × ?:λ ∥x∥ ≤ ? α}, where λ is a fixed positive scalar. In this case, we write (x ₀,f(x ₀))∈λ-extf. Here, we investigate the following question: if (x ₀,f(x ₀))∈λ-extf, wheref is a convex function, and if 〈f _n 〉 is a sequence of convex functions convergent tof in some sense, can (x ₀,f(x ₀)) be recovered as a limit of a sequence of points taken from λ-extf _n ? The convergence notions that we consider are the bounded Hausdorff convergence, Mosco convergence, and slice convergence, a new convergence notion that agrees with the Mosco convergence in the reflexive setting, but which, unlike the Mosco convergence, behaves well without reflexivity.     

Convex numerical radius

In this work we introduce the concept of convex numerical radius for a continuous and linear operator in a Banach space, which generalizes that of the classical numerical radius. Besides studying some of its properties, we give a version of James's sup theorem in terms of convex numerical radius attaining operators.     

凸对策核仁的非空性及合成凸对策核仁的单调性

本文给出了核仁与核及最小核心之间的关系 ,且证明了凸对策核仁的存在性和唯一性 ,证明了凸对策的合成对策仍是凸对策 .最后 ,我们讨论了合成凸对策的核仁不满足单调性 .     

Convex hulls in the hyperbolic space

We show that the convex hull of any N points in the hyperbolic space ${\mathbb{H}^{n}}$ is of volume smaller than ${\frac{2 (2 \sqrt \pi)^n}{\Gamma(\frac n 2)} N}$ , and that for any dimension n there exists a constant C _n > 0 such that for any set ${A \subset \mathbb{H}^{n}}$ , $$Vol(Conv(A_1)) \leq C_n Vol(A_1)$$ where A ₁ is the set of points of hyperbolic distance to A smaller than 1.     

Fermat's theorem as a corollary to the multinomial theorem

The multinomial theorem and a basic property of multinomial coefficients yield a way of counting the number of chains of coloured beads of a fixed length where each colour can be used repeatedly. This coupled with a simple counting argument forms a concise proof of Fermat's theorem.