The earliest sunset occurs about two weeks before the shortest day. Shouldn’t the shortest day also have the earliest sunset?
(and conversly with sunrise)
No, check into “analemma” and “equation of time”. The sun can be early or late by more than fifteen minutes–in other words, asymmetric about the time of noon. If you are offset in a time zone, at the far east side, or the far west side, the asymmetry can be even worse.
The two aren’t directly connected. The length of the day is dependant only on the orientation of the poles with respect to the sun. The shortest day (in the Norther Hemisphere) occurs when the north pole is pointed directly away from the Sun. The time of sunrise (or sunset) is determined by that AND latitude. In some locations, the time of earliest sunrise DOES correspond to the summer solstice.
This was discussed recently in this thread: http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=220240