In a script I am currently reading it is used at some point that the convex funciton $f = \ln (p(\exp (x_1 ), \ldots , \exp( x_n)))$ can be minimized in polynomial time under linear inequality constraints. Here $p$ is a polynomial in $n$ variables having nonnegative coefficients.
I have read multiple times before that a convex function can be minimized in polynomial time, but I am pretty sure that this blunt statement is not true in general, as you can produce some very ill behaved convex functions. So under what conditions can you minimize it in polynomial time, specifically what argument would be used for the function $f$ defined above? Is something like smoothness enough?