in this paper we obtain a necessary and a sufficient condition for the set of ω_0-nearest points ( ω_0-farthest points) to be non-empty or a singleton set in normed linear spaces. we shall find a necessary and a sufficient condition for an uniquely remotal set to be a singleton set.

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.

We study the possible Hausdorff limits of Julia sets and filled subsequences sequence dual Chebyshev polynomials a non-polar compact set $$K\subset {\mathbb C}$$ compare such to K. Moreover, we prove that measures maximal entropy for K converges weak* equilibrium measure on