Abstract: If A is a closed convex set in the complex plane then 9l(A; H) denotes all the normal (bounded linear) operators on the fixed separable Hilbert space H with spectrum contained in A. The fixed operator A has N as an 9t(A; //>approximant provided N belongs to 9L(A; H) and the operator norm \\A — N\\ equals pA(A), the distance from A to 9l(A; H). With some hypothesis on A, this note proves that the dimension of the convex set of all 9l(A; ir>approximants of normal operator A is (dim H^ where HQ is the orthogonal complement of ker(|^4 - F(A)\ — pA(A)) and F(z) is the unique distaince minimizing retract of the complex plane onto A. 1. Introduction. If A is a closed convex set in the complex plane then 9l(A; H) denotes all the normal (bounded linear) operators on the fixed separable Hilbert space H with spectrum contained in A. The fixed operator A has N as an 9l(A; /O-approximant provided N belongs to 9l(A; H) and the operator norm \\A - N\\ equals pA(A), the distance from A to 9l(A; H). The set of all 9l(A; -if")-approximants of A is denoted 9l(A; A). Provided A is normal and A is a closed convex subset of some straight line in the complex plane, the main theorem of this note constructs enough members of 91 (A; A) to determine its dimension as a convex set (see (7, pp. 7-9)). Thus, the main theorem generalizes previously known results for positive approximants (1, Theorem 5.2) and selfadjoint approximants (2, Corollary 3.3); it also answers analogous questions for best approximation by selfadjoint and nonnegative contractions. Halmos (4) suggests consideration of nonnegative contractions. It appears that the proof of the main theorem has isolated the precise ingredients for establishing such results. Perhaps greater clarity is a con- sequence. To eliminate trivialities it is assumed, henceforth, that A contains at least two points. 2. Preliminaries. Before the main theorem is proved, consideration is given to the hypothesis which is responsible for limiting A. The appropriateness of this hypothesis will be clear from the proof of the main theorem. If Q is some orthogonal projection on H, then QT\QH is the compression of F to QH.