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/