There are two kinds of convex grinds. One is done on a slack belt, one is flat ground on a platen with the near-edge area convexed. Seems like the slack belt grind is used more on large blades. Either type will produce more metal behind the edge for support... the slack belt type produces more metal all the way to the spine.
In convexing near the edge, I flat grind to a minimum edge, leaving some to avoid decarb at the edge and maybe avoid warping at the edge. Then, after HT, use stones to build the convex shape before finishing the entire blade.
Kevin Cashen said rececently, historical knives/swords have convex edge geometry... that is, do not have a secondary sharpening bevel. Wayne Goddard builds convex edges with a minor secondary bevel to facilitate resharpening by customers... a convex edge is harder for most folks to sharpen well, was my read on Wayne's statement.
Mike