Bregman distances without coercive condition: suns,Chebyshev sets and Klee sets

ABSTRACT

In a real Banach space X, we introduce for a non-empty set C in X the notion of suns in the sense of Bregman distances and show that C is such a sun if and only if C is convex. Also, we give some necessary and sufficient conditions for a compact set to be the Klee set, extending corresponding results on the Euclidean space.     

关于Chebyshev中心的一点注记

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Bregman distances and Chebyshev sets

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.     

Monic integer Chebyshev problem

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let denote the monic polynomials of degree with integer coefficients. A monic integer Chebyshev polynomial satisfies

and the monic integer Chebyshev constant is then defined by

This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied.

We compute for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases.

Conjecture. Suppose is an interval whose endpoints are consecutive Farey fractions. This is characterized by Then

This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater.

     

Sets with External Chebyshev Layer

Mathematical Notes -     

Proximality and Chebyshev sets

This paper is a companion to a lecture given at the Prague Spring School in Analysis in April 2006. It highlights four distinct variational methods of proving that a finite dimensional Chebyshev set is convex and hopes to inspire renewed work on the open question of whether every Chebyshev set in Hilbert space is convex.     

On feasible sets defined through Chebyshev approximation

LetZ be a compact set of the real space with at leastn + 2 points;f,h1,h2:Z continuous functions,h1,h2 strictly positive andP(x,z),x(x ₀,...,x _n ) ⁿ⁺¹,z , a polynomial of degree at mostn. Consider a feasible setM {x ⁿ⁺¹z Z, –h ₂(z) P(x, z)–f(z)h ₁(z)}. Here it is proved the null vector 0 of ⁿ⁺¹ belongs to the compact convex hull of the gradients ± (1,z,...,z ⁿ ), wherez Z are the index points in which the constraint functions are active for a givenx* M, if and only ifM is a singleton.This work was partially supported by CONACYT-MEXICO.     

Julia集和等势线上的Chebyshev多项式

用P表示一个度为d的首一多项式,J_P表示它的Julia集.本文得到Julia集J_P和其等势线Γ_P(R)上的d~n-阶Chebyshev多项式,并举例说明二者并不总是相等.     

The integer Chebyshev problem

We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases.

We examine the case of the interval in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise.

We also examine some of the structure of such minimal ``integer Chebyshev' polynomials, showing for example that on small intevals and for small degrees , achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys and others, as to what the ``integer transfinite diameter' of should be. We show that this conjecture is false.

The problem is then related to a trace problem for totally positive algebraic integers due to Schur and Siegel. Several open problems are raised.

     

Characterisations of Chebyshev sets in c0

A subset MX of a normed linear space X is a Chebyshev set if, for every xX, the set of all nearest points from M to x is a singleton. We obtain a geometrical characterisation of approximatively compact Chebyshev sets in c₀. Also, given an approximatively compact Chebyshev set M in c₀ and a coordinate affine subspace Hc₀ of finite codimension, if M∩H≠, then M∩H is a Chebyshev set in H, where the norm on H is induced from c₀.     

The relative Chebyshev centers of finite sets in geodesic spaces

In the present paper we estimate variation in the relative Chebyshev radius R _W (M), where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We find the closure and the interior of the set of all N-nets each of which contains its unique relative Chebyshev center, in the set of all N-nets of a special geodesic space endowed by the Hausdorff metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this set.     

Methods of Chebyshev points of convex sets and their applications

Chebyshev points of bounded convex sets, search algorithms for them, and various applications to convex programming are considered for simple approximations of reachable sets, optimal control, global optimization of additive functions on convex polyhedra, and integer programming. The problem of searching for Chebyshev points in multicriteria models of development and operation of electric power systems is considered.     

A Chebyshev type inequality for Sugeno integral and comonotonicity

We supply a characterization of comonotonicity property by a Chebyshev type inequality for Sugeno integral.     

Chebyshev type inequality for Choquet integral and comonotonicity

We supply a Chebyshev type inequality for Choquet integral and link this inequality with comonotonicity.     

Approximating weak Chebyshev subspaces by Chebyshev subspaces

We examine to what extent finite-dimensional spaces defined on locally compact subsets of the line and possessing various weak Chebyshev properties (involving sign changes, zeros, alternation of best approximations, and peak points) can be uniformly approximated by a sequence of spaces having related properties.     

The Chebyshev centers in a normed linear space

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ON THE GENERALIZATIONS OF KALANDIA＇S LEMMA

Based on Bernstein＇s Theorem, Kalandia＇s Lemma describes the error estimate and the smoothness of the remainder under the second part of Hoelder norm when a HSlder function is approximated by its best polynomial approximation. In this paper, Kalandia＇s Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Hoeder norm or the weighted HSlder norms.     

Some new Chebyshev spaces

In this note we see another circumstance where Chebyshev polynomials play a significant role. In particular, we present some new extended Chebyshev spaces that arise in the asymptotic stability of the zero solution of first order linear delay differential equations with m commensurate delays where a_j,j=0,…,m, are constants and τ>0 is constant.