Aequat. Math. c  Springer International Publishing 2016 DOI 10.1007/s00010-016-0405-3 Aequationes Mathematicae Best constant in stability of some positive linear operators Dorian Popa and Ioan Ras ¸a Abstract. We prove that the kernels of Bernstein, Stancu and Kantorovich operators are proximinal sets, therefore the infimum of Hyers–Ulam constants is also a Hyers–Ulam con- stant for the above mentioned operators. Moreover, we investigate what happens when the supremum norm is replaced by the L 1 -norm. Mathematics Subject Classification. 39B82, 41A35, 41A44. Keywords. Hyers–Ulam stability, best constant, proximinal set. 1. Introduction Hyers–Ulam stability is one of the main topics in functional equation theory. Ulam formulated a problem concerning the stability of the equation of ho- momorphism of a metric group in 1940 and a year later D.H. Hyers gave a first answer to Ulam’s problem for the Cauchy functional equation in Banach spaces. Recall the result of Hyers [9]: Let X, Y be two real Banach spaces and ε> 0. Then for every mapping f : X → Y satisfying ‖f (x + y) − f (x) − f (y)‖≤ ε, x, y ∈ X, (1.1) there exists a unique additive mapping g : X → Y such that ‖f (x) − g(x)‖≤ ε, x ∈ X. (1.2) This is the reason why this type of stability is called after their names [9, 23]. The result of Hyers was extended later by Aoki [2] and Rassias [20] by re- placing ε in (1.1) with a function depending on x and y. Generally, we say that an equation is stable in the Hyers–Ulam sense if for every solution of a perturbation of the equation (approximate solution) there exists a solution of the equation (exact solution) near it. Hyers–Ulam stability was considered by numerous mathematicians, especially during the last 50 years, due to its