Be careful on that "turn in" idea. That works both ways and it also doesn't work as you may think.
If the edge of a concave catches first on a severe angle hit, the blade may indeed try to "turn in" but that will throw the entire arrow off its course and you will dump and lose nearly all of its penetration power. The other scenario is that it travels straight on and creates a long rip in the skin of the critter, maybe only skin deep.
Conversely, if a very convex blade makes the same hit, it may very well do the same ripping as the concave, or simply bounce away from the critter. I have seen this in action myself.
That is something to consider when choosing such heads, that it may limit the shot angle you should take.
Both convex and concave designs ( as well as straight heads) can make wonderful, deadly broadheads.
ChuckC