Data publishing under privacy constraints can be achieved with mechanisms that add randomness to data points when released to an untrusted party, thereby decreasing the data's utility. In this paper, we analyze this privacy-utility tradeoff for the pointwise maximal leakage (PML) privacy measure and provide optimal privacy mechanisms for a general class of convex utility functions. PML was recently proposed as an operationally meaningful privacy measure based on two equivalent threat models: An adversary guessing a randomized function and an adversary aiming to maximize a general gain function. We prove a cardinality bound, showing that output alphabets of optimal mechanisms in this context need not to be larger than the size of their inputs. Then, we characterize the optimization region as a (convex) polytope. We derive closed-form optimal privacy mechanisms for arbitrary priors in the high privacy regime (when the privacy parameter is sufficiently small) and uniform priors for all ranges of the privacy parameter using tools from convex analysis. Furthermore, we present a linear program that can compute optimal mechanisms for PML in a general setting. We conclude by demonstrating the performance of the closed-form mechanisms through numerical simulations.