I recently came across this paper:
which has nice formula for a “generalized unit ball” in :
These balls can look pretty crazy (as some pictures in the paper show).
The main result is that for , the volume is equal to
The formula for the volume of the -sphere in the norm is well known, but this formula lets us calculate all sorts of volumes. For example, for the unit ball we get the rather clean and beautiful formula
The proof given in the note is by induction, and a remark at the end points to several other proofs based on Laplace transforms.