A lenticular shape consisting of the intersection of two slightly-overlapping spheres, or a sphere and a half-space (the part cut off a sphere by a flat plane, a plano-convex lens) can be lapped smooth. So can the difference of a sphere (or such a lenticular shape) and a cylinder running through its center; so can a cylinder with two flat faces. These shapes, of which the lenticular shapes are the simplest, have rotational symmetry around a single axis of rotation. So by supporting one of them on an air bearing you can get an air bearing that resists movement in five degrees of freedom, rather than the three or four you get from a planar or endless-cylinder air bearing.
(In practice you probably want to trim the edges of the lens short of being a true knife edge.)
If you’re using bearings in pairs rigidly joined by a shaft, as is normal practice, you don’t need the lenticular shape; you can just use a sphere. Four bearing pads on a sphere (or lens) are sufficient to support it, but two bearing pads on each of two spheres joined by a rigid shaft — either all four compressing the shaft or all four stretching it — would also be enough. In many applications it would be desirable to tolerate a little axial misalignment in this way. In applications where the axial misalignment is accompanied by uncontrolled axial tension or compression, you might want to use four pads per sphere anyway, just in case the shaft gets put into tension when compression was expected, or vice versa.
It is not necessary that the shaft joining the two spherical sections be coaxial; the axis of rotation will run accurately through the centers of the spheres regardless of where the shaft is. Indeed, the "shaft” could even be a C-shaped thing that goes around the outside of the two trailer park girls. Uh, the two balls.