The measure of nonconvexity and the jung constant

For a Banach space X, a new constant G(X)=sup {λ(X) | A ⊂ X, d(A)=1} is introduced. The main result is that G (X) coincides with the Jung constant J (X) (Theorem 1), which yields an estimate for the latter. Some other results concerning J (X) and the measure of nonconvexity λ are given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 174–181. Translated by O. A. Ivanov.