The Senate is currently controlled by the Republicans, but control might shift to the Dems with the election results (Ds are poised to pick up seats in Alaska, Oklahoma, and possibly Kentucky, of all places). This would set the following rushed timetable off, hypothetically: the R's get their Supreme Court justice to resign right away if ill. A nominee is named and the Senate skips confirmation hearings and gets right to the votes. This would set up the mother of all fillibusters by the D's to prevent the justice from being confirmed before the January 2nd swearing-in of the new Senate. Whatever case the Supreme Court has would be heard, but might be voted on 4-4, in which case the nature of the precedent being requested by the appellate would determine the disposition of the case. (There are no tiebreakers in the Supreme Court; this is why, despite the obvious conflicts of interest of at least two of the Supreme Court justices last time, Scalia and Thomas, in having family members in the employ of the Bush campaign, there were no recusals.)
How then would the victor of the election be determined? I have no idea, but you can guarantee that it will be extra-constitutional.