Challenges of round doors

There is, apparently, a construction company out there that will build you a house styled on a Hobbit hole. Naturally, an important question arises for Jewish owners of such homes: do you put a mezuzah on a round door, and if so where? The mezuzah is the scroll (containing certain torah passages) in a case that is affixed to your doorpost -- so what exactly is a doorpost?

The door has to open and close, but it seems possible to design one that has only enough straight section to support a super-strong hinge. Relevant questions seem to be: (a) how long a straight section do you need to qualify as a doorpost, and (b) does your doorpost have to go all the way to the ground? (Is it still a "post" if it just kind of floats there?)

(An answer cites sources that a round door does not require a mezuzah.)