Hyers-Ulam stability of isometries on bounded domains

More than 20 years after Fickett attempted to prove the Hyers-Ulam stability of isometries defined on bounded subsets of R n {{\mathbb{R}}}^{n} in 1981, Alestalo et al. [Isometric approximation, Israel J. Math. 125 (2001), 61–82] and Väisälä [Isometric approximation property in Euclidean spaces, Israel J. Math. 128 (2002), 127] improved Fickett’s theorem significantly. In this paper, we will improve Fickett’s theorem by proving the Hyers-Ulam stability of isometries defined on bounded subsets of R n {{\mathbb{R}}}^{n} using a more intuitive and more efficient approach that differs greatly from the methods used by Alestalo et al. and Väisälä.