The best constant in the Hyers-Ulam theorem on isometric approximation in Hilbert spaces is equal to the Jung constant of the space.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-1-3, author = {Taneli Huuskonen and Jussi V\"a\i s\"al\"a}, title = {Hyers-Ulam constants of Hilbert spaces}, journal = {Studia Mathematica}, volume = {151}, year = {2002}, pages = {31-40}, zbl = {1016.46022}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-1-3} }
Taneli Huuskonen; Jussi Väısälä. Hyers-Ulam constants of Hilbert spaces. Studia Mathematica, Tome 151 (2002) pp. 31-40. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-1-3/