I was going to call this the convex hull of the usage-efficiency relationship, but decided that would truly scare away everyone who was actually going to be interested in this.
You have probably heard of the usage-efficiency relationship. It's debatable in some circles, and taken for granted in others. The idea is that the more a player shoots, in general, the less efficient he becomes. Ideally, we would just regress TS% (or some other measure of efficiency) onto USG% (or some other measure of volume) and see a nice, tight (well-correlated) and linear relationship. Like this one for 2012 (data points are for players with greater than 500 possessions):
Huh. That looks pretty crappy and not at all what we were expecting to see, right? Well, one complicating factor is that some players that don't shoot a lot sometimes are very efficient (see Tyson Chandler) while other players that don't shoot a lot are very inefficient (see Ben Wallace). This complicates the matter. Others have thought of ways to tease out the hypothesized effect (see Eli Witus' great study for one of the best examples).
I kind of stumbled upon another way to possibly see this effect. I was looking at players (since the 3-pt shot was installed) that have shot greater than 55% TS and had greater than 25% USG. I took these 300 or so players and plotted TS vs. USG and this is what I found:
A convex hull is the set of points that lies at the boundary of a N-dimensional function. In 1-D, it would simply be considered the two end points of a line. When we plot TS% vs. USG% for these top players, what we find is that there is a boundary running along the diagonal that corresponds very well with our intuition underlying the debate. And although I can't prove it, I think this is a real effect. And it's amazingly linear!
It's also interesting to see the names along that boundary. They are not unexpected, are they? Now, getting back to 2012. There are two players who come close, but fall short of greatness (defined in this way, at least):