On the rate of convergence of certain approximation operators.
We will generalize and improve the results of T. Singh [Publ. Math. Debrecen 40 (1992), 261-271] obtaining the L. Leindler type estimates from [Acta Math. Hungar. 104 (2004), 105-113].
Let X be a normed space. A set A ⊆ X is approximately convexif d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with and , where ℋ denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0, we construct worst possible approximately convex sets in C[0,1] such that ℋ(A,Co(A))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.
This survey is a tribute to Géza Grünwald and Józef Marcinkiewicz dealing with the so called Grünwald-Marcinkiewicz Theorem.