How much does the sound definition vary during an LP (Vinyl)? This question came to me when I realized how the linear speed varies while listening to a Vinyl LP.
The linear speed variation has to be compensated with a variation in the resolution of the grooves, that is, since the linear speed decreases, the groove resolution also has to decrease in some measure. What is this measure of reduction, or else, how much does the linear speed reduce? And how does that influence the sound definition?
 A: It should be obvious that the linear speed is inversely proportional to the distance from the center to that point on that groove.
Yes, speed varies over a record, so the wavelength of the wiggles in the groove gets shorter for the same frequency as you get further into the record.  However, the master record was created with this same phenomenon in reverse, so it all works out.
The resolution that the vynil can be pressed with exceeds what is required even for encoding the high frequencies near the end of the track (closest to the center).
Since dust and dirt particles will be the same size regardless of where they land on the record, the noise from this dirt will have a overall lower spectrum at the end of the track than the beginning.  Since the standard RIAA playback profile is heavily low pass filtered, this argues that there should be on average relatively more noise from dirt at the end of a track.  There are other sources of noise too that don't scale the same way.
Overall, the fact that linear speed changes over 2:1 from the beginning to the end of a track is one of the engineering tradeoffs that went into records.  This is also true of floppy disks and hard drives.  It is interesting to note that it is not true for CDs.  The CD standard specifies constant linear track speed for the same data rate.  CD drives (and their decendents like DVD drives) have variable speed motors that compensate for the diameter effect to keep linear track speed constant.
A: Your question is a little misstated, since it's not so much the "groove resolution" as the high linear frequency limit to which the groove can be cut.  The stylus (and piezo or magnetic components which convert motion to electrical signal) doesn't really care about linear speed in the along-track direction, just the transverse speed.  So long as the groove walls can be cut such that, e.g.,  15kHz undulations are faithfully reproduced in the vinyl, the sound will be fine.
There are other problems, including "hiss" and physical damage to the outer grooves where the stylus is moving longitudinally much faster than in the inner grooves, and (depending on just how fancy your turntable is) cross-track skate force and rotational misalignment of the stylus' axes to the groove direction.  
