We present some sufficient conditions ensuring the upper semicontinuity and the continuity of the Bregman projection operator Π g C and the relative projection operator P g C in terms of the D-approximate (weak) compactness for a nonempty closed set C in a Banach space X . We next present certain sufficient conditions as well as equivalent conditions for the convexity of a Chebyshev subset of a...

We study the relationship between transfinite diameter, Chebyshev constant and Wiener energy in the abstract linear potential analytic setting pioneered by Choquet, Fuglede and Ohtsuka. It turns out that, whenever the potential theoretic kernel has the maximum principle, then all these quantities are equal for all compact sets. For continuous kernels even the converse statement is true: if the ...

We prove that in a convex metric space (X, d), an existence set K having a lower semi continuous metric projection is a δ-sun and in a complete M -space, a Chebyshev set K with a continuous metric projection is a γ-sun as well as almost convex.

Two types of stability of the lexicographic set for the multicri-teria problem on a system of subsets of a finite set with the vector criterion of the most general kind are investigated. Lower bounds of stability radii have been found for the case where Chebyshev norm was defined in the space of vector criterion parameters.

Let f : z 7→ z + c be a quadratic polynomial whose Julia set J is locallyconnected. We prove that the Brolin measure of the set of biaccessible points in J is zero except when f(z) = z − 2 is the Chebyshev quadratic polynomial for which the corresponding measure is one. §

Let X be a non-reflexive real Banach space. Then for each norm | · | from a dense set of equivalent norms on X (in the metric of uniform convergence on the unit ball of X), there exists a three-point set that has no Chebyshev center in (X, | · |). This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.

We consider a compact set K in the form of a sequence of segments. By means of nearly Chebyshev polynomials for K, the modulus of continuity of the Green functions gC\K is estimated. Markov’s constants of the corresponding set are evaluated.

The eponymous theorem of P.L. Wantzel [Wan] presents a necessary and sufficient criterion for angle trisectability in terms of the third Chebyshev polynomial T3, thus making it easy to prove that there exist non-trisectable angles. We generalize this theorem to the case of all Chebyshev polynomials Tm (Corollary 1.4.1). We also study the set m-Sect consisting of all cosines of m-sectable angles...

In this paper, the dynamics of the Chebyshev-Halley family is studied on quadratic polynomials. A singular set, that we call cat set, appears in the parameter space associated to the family. This cat set has interesting similarities with the Mandelbrot set. The parameters space has allowed us to find different elements of the family such that can not converge to any root of the polynomial, sinc...