Measuring to the 3rd decimal on a vernier caliper is what you need to refer to , as .. if you use a more casual tool it will give you too generalized an answer.. 27.200mm ideally though 27.195 would be close , as would 27.215 , for example
precision tools give you precise answers.

and if its a Carbon post , I'd go with the carbon assembly paste ,
If you are a heavy rider , then I'd look into a quality Aluminum seat-post instead ,
then the normal wipe of grease is adequate.
Re do that regularly to keep post from getting stuck.