What is the exact meaning of this integral on this page? I can understand that we are integrating the proportion over solid angle, but I don't think this result is the final proportion of L_o comparing to L_i (since L_i varies between solid angles)
I found that on wikipedia the integration is on w_o not w_i, which makes more sense. I think you made a typo here.
It makes sense that Helmholtz reciprocity should generally hold, but is it a physical truth across all surfaces? A quick Google search seems to indicate that perhaps there are some cases in which it does not, though there aren't a lot of details about why these cases cause it to be violated. (For example, the Wikipedia article makes an allusion to a magnetic field causing the optical medium to "depart from linearity", though I don't really follow what it's talking about.)
To my understanding, only for glass surface, the distribution function would equal to one, otherwise, there would be energy reflected to other directions.
@yongchi1 Yes, thanks; that's a typo indeed.
@mdsavage Right; a general rule about physics is that you can always cook up exotic scenarios where some assumption doesn't hold. There's a very nice discussion of Helmholtz reciprocity, as well as a good list of references, in Chapter 6 of Eric Veach's (Academy Award-winning) PhD thesis.
@riceroll If you correct the typo mentioned above, then you can have a material that reflects 100% of the light, but not necessarily in the specular direction.