In this paper several results on certain constants related to the notion of Chebyshev radius are obtained. It is shown in the first part that the Jung constant of a finite-codimensional subspace of a space C(T) is 2, where T is a compact Hausdorff space which is not extremally disconnected. Several consequences are stated, e.g. the fact that every linear projection from a space C(T), T a perfect compact Hausdorff space, onto a finite-codimensional proper subspace has norm at least 2. The second discusses mainly the "self-Jung constant" which measures "uniform normal structure." It is shown that this constant, unlike Jung’s constant, is essentially determined by the finite subsets of the space.