My estimation for my best freehand edges is approx 1/2 a degree from apex to shoulder, most are likely closer to a degree or just under, but just a guess. I suspect this holds true on V bevel and Scandi (giving the Scandi an advantage in overall "flatness" as that 1/2 degree is spread out over a larger area), my full convex edges could really only be judged by how consistent the apex angle stays after multiple sharpening. I'm not sure if this works out to .001 though, but maybe...a single sheet of writing paper is approx .0035, and copy paper maybe .0045 - deviation equivalent to 25% the thickness of a sheet of paper. Is quite possible.
Let's for example look at an edge bevel that is .040" thick at the shoulder, and bevel surfaces that are .060" wide. This would make an edge angle of about 19.5 degrees. Now if we varied that angle up to 19.97 degrees, then the thickness at the shoulder becomes .041. We can't really calculate the thickness at the edge ( well, not ideally anyway ) so we're really more concerned with this thickness at the bevel shoulder, and the thickness mid-way down the bevel face to the cutting edge. If you have the variation in the angle to produce that .041 increase in width, then instead of a perfectly flat plane there could be a curve that gradually widens from somewhere close to .040" to .041" near the middle, and then thinning back down again toward the edge, thus producing the convex nature.
Now using a lot of trig for other edge widths and bevels you can see smaller and larger variations are possible. Take a scandi grind for example. I have one with an thickness of .098", and a bevel width of .210" giving it an angle of 26.9 inclusive. Say I wavered a little and now the thickness is closer to .096 at the bevel shoulder. That angle would still be 26.5 degrees. If the bevel thickness was half of that, say .105 then the difference would be between 27.8 and 27.2 degrees. So with the wider bevel surface, the variation in edge thickness throws the angle off by .4 degrees, but by .6 degrees when it is only half as wide. Given that, and with those specific geometries, one could expect to see differences in flatness of +/- .002" with under 1 degree of angle deviation.
So given all this, if you work in larger variations than 1/2 a degree ( which honeslty for most hand shaperners will probably be the case, and then depending a lot on how wide a blade one is sharpening ) then the variation in edge thickness (and thus flatness) will be greater as well. I'm guessing it could vary as much as +/- .005" or greater depending on various edge geometries. A good question is whether this amount of variance is even discernable by eye; going by my experience I'm going to say just barely, and that edges that seem "perfectly flat" to the naked eye still maintain at least .001" of variance. Now is this really the same as being a convex edge? I do not think so, it is just simply being "not flat". Because there is nothing to dictate that in this +/- .005" variation that there are not other flat spots, or that the shape itself is actually useful in anyway. Instead of having one contiuous arc, you could have many different bevels but with blended shoulders to appear like one continuous arc.
With that in mind, that is actually how I would grind precision radiuses ( radii? ) onto machine parts with hand files. You begin filing in many bevels at different angles, and then rounding them out with sand paper using a radius gauge to check. When doing this it is important to get the right radius and one can't simply make it "close" to being the right curve or the part won't work for the machine. In this respect I do not think there are many people who grind in a convex edge doing this, frankly because I don't think having a specific size of radius would be useful, and because the radii on convex edges would actually be so large and the surface so small, that checking with any kind of gauge wouldn't really be possible. We're talking about radii over 1", but on surfaces whose width are less than 1/10th" an inch. Perhaps some type of laser raidus measuring devices exist but I don't believe a person could use any hand-tool to measure this without using trig to figure out what angles you'd need to blend together as we did in the machine shop.
However, relating back to how I filed in radii, if using a knife blade one first grinds in one bevel of one angle, then one of a slightly larger angle, then of a slighthly larger angle yet and then rounds out these bevel surfaces into one contiuous surface, this is probably the closest thing you can get to a precisely made convex edge. I'm sure someone could figure out what size bevels and at what angles to grind them to approach a certain radius for the surface because that's how I did it (under the guide of a much smarter instructor mind you), but the real question is whether it would actually be applicable and as I described earlier I don't think it really would be, just based on the level of skill people would need to reproduce these shapes. Not to boast but we were being trained by a guy that worked at NASA an insisted we hand-machine parts to within +/- .005" tolerances. I've actually made many angle gauges that have 1/2 degree tolerances using only a hacksaw and a handfile--this was required before we could use the actual machines.
In all actuality, having a convex surface that is perfectly circular is not even how they are always ground. Some have more swell approaching the cutting edge, some have more swell in the back, so instead of forcing the curvature of a radius they form shapes more commonly comparable to "apple seeds" or "tear drops" that are actually comprised of one radius intersecting another raidus, if you want to get technical and think of a way to measure such shapes. The difference in performance these shapes have on
edge grinds is probably really hard to discern, but when talking about full blade grinds in these different shapes it makes a huge difference for things like swords, choppers and filet knives to have a slight curvature, however I don't think having that curve being perfectly circular or to a perfefectly measurable radius will ever be useful except for reproduction purposes and even then the type of precision machining/crafting needed to do so would be so intensive, the price would be far too high to merit the actual difference in performance. So in other words, there's never going to be a filet knife mass produced with a convex edge that has a radius of a certain amount, that transitions into a different radius to form a specific shape--best bet would they would simply grind on a 10" or more radius as the "convex" shape. Pretty much everything else is more or less art, with the craftsman forming the shape more by intuition and experience than by numbers and math.
I put convex grinds on my kitchen knives but it is hard to find the right kind of curvature to actually benefit. I always start out making three bevels of different angles, and then blend them together. One needs to have a strong visual imagination to be able to do this and produce a convex shape that is actually intentional and offers a performance gain--I always draw it out on paper. "Incidental" or "accidental" convexing is not the same as intending to make a blade form a certain shape that behaves as intended. I find convex blades that have more swell toward the cutting edge split better, such as an axe head or cleaver. A swell in the middle parts material and reduces friction better, like a filet knife or a katana. Typically the swell being closer to the edge shoulder or spine doesn't really do anything in my experience, I suppose theoretically it could reduce drag on the edge-bevel's shoulder, but only if we're talking edge grinds and not blade grinds. As I said before, all of this makes more of a difference for an entire blade's grind than it does for the edge grind, as these shapes don't actually cause a real discernable difference until the dimensions are large enough that control is actually effected through various materials.
However all that being said, I'll say again I don't think there is a large performance difference between a V grind that is exceptionally flat--even one done by a guided machine--and an "incidentally convexed" hand-sharpened edge. Edges that excel because of their shape have to be designed that way, it doesn't often happen "accidentally". Whether it is perfectly flat or not, is not what makes the difference, it's how flat and where it is flat.
I think convex edges give the perception of being easier to sharpen mainly because there is a notion with a V ground edge that if you do not maintain the flatness of the bevel, that you are altering the angle and "not doing it right" or not making a "true V grind". When people sharpen a convex grind, the change in shape is far less noticeable for one thing so the results seem much more satisfactory, but really I think it lies in just how difficult it actually is to keep a V grind very flat. It adds to some of the perceived difficulty of it. This is why a lot of the time, instead of calling it "convexed", people simply refer to them as "free-handed" or "sloppy" and the craziest part is that they're generally just as sharp as they would be if they were ground in an exceptioanlly flat V shape. Of course some don't feel quite the same so prefer guided options--I suppose that still comes down to a matter of opinion.
One thing that really aggravates me is when people suggest it is not a "true Scandi grind unless it is compeltely flat". Then they go on to give such loose defintions of having a "large bevel". I doubt Carter is really trying to cash in on the "convex" gimmick as much as he is trying to point out that for 90% of the knives and edges people are sharpening, whether it is convex or flat does not make a discernable difference in performance, and trying to guide people away from hard-and-fast rules such as "If it is not perfectly flat, it's not a V bevel," or the notion that an edge even need be extremely flat
or convexed to perform well, but honestly I didn't watch the video so can't comment on that specifically.
But yeah... There's my $.02... Or more like my $1.50