If you find yourself thinking about the differences between geometric expected utility and expected utility in terms of utility functions, remind yourself that, for any utility function, one can choose* either* averaging method.
No, you can only use the geometric expected utility for nonnegative utility functions.
No, you can only use the geometric expected utility for nonnegative utility functions.
If bounded below, you can just shift up to make it positive. But the geometric expected utility order is not preserved under shifts.
Thanks for pointing this out, I missed a word. I have added it now.