I can't tell whether you're talking about blade grinds or edge grinds or both. I'll guess edge.
The problem in comparing convex to V edges is that people don't define the edges with actual geometry. A V edge can have more or less metal behind the edge than a convex edge, depending on how they are ground.
But in general, when compared fairly -- same shoulder width and height -- a V edge will cut better because it has a more acute angle and a convex edge will be more robust because it has more metal behind the edge.
I agree with this statement fully, but one must point out that at the apex, the convex edge will have a much greater angle than the V edge, hence the edge will not be as keen for cutting purposes (might be repeating what you said). If I develop a convex edge by sharpening equal width facets from 15, 16, 17 and 18 dps with 18 degrees reaching the apex on both sides, then strop the edge to remove the facets forming a radius convex geometry, this knife will have less metal behind the edge, and hence be weaker, than if I sharpen the knife in a V grind all the way to the apex at 18 dps. In this scenario, the convex edge will likely cut more efficiently than the V grind because of the reduced metal behind the edge, but it will be weaker. The scenario that you described above is if I convexed the edge as stated, then compared it to a V grind all the way to the apex at 15 dps. The latter would cut better and be weaker.
However, I think that the OP was talking about convex/hollow/flat grinds. In all of these grind scenarios, you get into thicker metal with repeated sharpening's of a knife, as you are moving the edge up the blade. In order to maintain or achieve suitable blade geometry for the life of the blade, this grind must be thinned down in order for proper edge geometry and cutting performance. If I would rank the three grind types from least to most with respect to the amount of metal that must be removed to maintain proper edge geometry and cutting efficiency, it would be hollow (least) followed by flat and a convex grind would require removing the most metal in order to maintain proper cutting. Thinning hollow ground knives and flat ground knives are relatively easy by simply placing the grinds flat on a stone and grinding it while maintaining the original grind angle. On a hollow ground knife, this will form two train track like grind patterns, one near the edge and one near the flat. With repeated sharpening and grindings, these tracks will continue to get wider until they converge in the middle, hence making a full flat grind. Thinning flat ground blades will remove metal across the entire grind, also very easy to do on a flat stone. Convex grinds, on the other hand, are nearly impossible to do on a flat stone. One would have to resort to taking the blade to a belt sander with a spongy backing behind the belt to effectively thin this blade while maintaining the intended geometry. A very careful and skilled hand would be required.
In use, a convex ground knife is less likely to stick in wood when performing chopping or splitting duties due to the radius of the grind (think axe), but the maintenance problems (for me at least) in maintaining proper edge geometry make me avoid this grind. Other than this, I see no benefit to a knife with a convex grind.
CRK will regrind the blades of his knives for free if the knife is sharpened to the point that the geometry is off. Personally, I cannot see myself grinding the sides of an $800 Hinderer each time I sharpen it, or even a CRK. But I use this method for all of my other knives at each sharpening. This is another reason why I do not like coated blades. I am going to be grinding the coating off at the first sharpening.