I am wondering about the relationship between the Krein-Milman theorem (KM) and some other weak forms of the axiom of choice (AC). I currently basically know the following:
KM + BPI (Boolean prime ideal theorem) implies AC. (Bell, Fremlin, A geometric form of the axiom of choice)
I also know the following:
DC (dependent choice) + BPI does not imply AC (consequently DC does not imply KM). (Pincus, Adding Dependent Choice to the Prime Ideal Theorem)
I have two closely related questions about these axioms, but have been unable to find an answer. These are
Does KM imply DC?
Is KM consistent with AD (axiom of determinacy)?
If we replace ''KM'' with ''DC'' in the above questions, the answer is always "yes" (for question 1, this is of course trivial). So my underlying motivation is to figure out how similar KM and DC are.