Something that was very handy during working out the 2D version of the algorithm was that I could sweep around a circle to generate an approximate set of points on the Minkowski sum and draw it as a polygon, so I could then draw the simplex I was refining.
So the first step for implementing a 3D version has been to create a method that can generate a mesh from a shape given only a support function. It works by sweeping directions around a unit sphere to generate the vertices, then stitching these into triangles, taking advantage of the fact that we know the order the vertices were added and rejecting any degenerate triangles that are formed.
Above is an image of a capsule built by combining two spheres, using the maximum dot of the point with the direction of each in the support function, but the mesh generator can work with any shape that you provide a support function for, so will be ideal for also visualising the Minkowski sum.
It creates a few more triangles than are actually needed so would require some refinement if it was to be used as anything other than a visualisation tool but is fine for these purposes.