Height of doorposts:
Each of the two doorposts must reach a height of at least ten Tefach to be considered a Halachic doorpost which requires a Mezuzah.
Width of the doorway-Must the entrance of the doorway contain a minimum width of opening?
Some Poskim rule the doorway does not contain a minimum width, and hence so long as one can pass through the opening it is obligated to have a Mezuzah. Other Poskim however rule that the doorway must have a minimum opening of 4 Tefach [32 cm]. [Practically, if the opening is less than 4 Tefachim one is to place a Mezuzah without a blessing. If however the opening is less than three Tefach [24 cm] then it is exempt from a Mezuzah according to all.]
 Rama 287/2; Michaber 286/2 regarding a domed entrance.
 Bear Hagolah 287; Maggid Mishneh Shabbos 16/20 in understanding of Rambam which does not mention a minimum width.
 Shach 287/2 as rules Rashi; Tur; Mordechai
 Aruch Hashulchan 287/9
 Aruch Hashulchan 287/9; Pischeiy Shearim 287/58
The reason: As less than 3 Tefach is considered Lavud, and is hence considered as if it is completely closed. [ibid]