Well, I was half hoping this topic would just die off, but now I can't seem to keep away from it.
Quick survey:
What kind of thickness and angles are folks actually thinking about here when discussing this topic? How many of you regularly carry & use blades with an edge formed at 5-7 degrees per side, with a thickness above the edge bevel of .006" or less? (Great if you do!) How many finish up their sharpening with the angle at the very edge more like 10*-15* per side?
I think a good reason for the convex edge's popularity is that factory made knives come with edges that are too thick 99% of the time. This is one way to thin things out where the edge bevel and primary grind meet, without having to regrind the entire blade.
In the South, something called whittling is a favorite pastime in small towns. The best grind for this is either a full hollow, with a very shallow "V" edge, or a true Flat Grind where the flats extend all the way to the edge. Similar to a Japanese Swords edge. This is sometimes called a Zero grind. Many old timers will lay a blade flat on a stone to sharpen it. These knives are like razors. If you know anything about whittling, or carving, you know that you can't run a curl with a convex edge. it will skip out of the cut, unless you use a very steep angle in the cut. Then it will simply dig in. Ruining the cut.
Mike, this contradicts much of my own experience as well as stuff I've seen from others on the subject. First off, within reason none of these edges should have any trouble whittling a fine curl as long as they're sharp. They will all have a little bit of trouble coming back up out of the wood (depending on wood type) on deeper concave cuts. But convex will do a better job. Contrast it with a full flat grind/zero edge. As the blade is scooping a curve into and back out of the wood, the only parts of the blade contacting the wood will be the very edge and spine. As you try to twist/guide the edge back up & out, you are using the spine as a leverage point against the bottom of the cut. Since the spine is so far away from the edge, you have less control. With a full convex grind, you can lever against the sides of the blade, which are much closer to the edge.
I have seen this exact phenomenon spelled out clearly with several illustrations in a big wood working book. I've seen it myself in action in all kinds of applications. Like when I'm carving wood with a draw knife- use the flat side for making straight cuts, and flip it over to the "wrong" side for better control on concave curving cuts. Or clearing brush with a machete- I make sure to sharpen the edge with a flat V, so the blade will continue cutting on course as the sapling bends. Whereas a convex edged machete will tend to jump right back out of the cut, and directly back at my legs

eek

if the angle of attack is too steep. Or even on the bucket of our loader tractor- Dad welded an old grader blade to the front wrong way up, and now it only wants to dig deeper. I've seen Japanese sword guys discuss this in regards to cutting tatami.
Regarding the swords- Maybe I misunderstood you; were you only referring to the fact katana don't have a secondary edge bevel? Because while there are all kinds of obscure grinds used on Japanese swords, by far the most common is basically a full
convex grind, with a relieved spine area. Not a flat V like is more common on modern competition swords.