On the Convexity of the Upper Bild

We give a proof of the convexity of the upper bild of a quaternionic matrix, a theorem of So and Thompson. After reduction to $2\times 2$ matrices, we introduce real coordinates on a sphere and obtain two representations of the upper bild over the closed unit ball, involving a convex function and a concave function. Minimizing the first and maximizing the second over parameters with a fixed real part yield a convex lower boundary function and a concave upper boundary function, from which convexity follows.