Convex maximization encompasses a broad class of optimization problems and is generally $\mathsf{NP}$-hard, even for low-rank objectives. While polynomial solvability has been established for several important special cases, a broader structural understanding for mixed-integer settings remains limited. In this paper, we study this question from a geometric perspective and identify a structural property of the feasible region underlying tractability, which we term \emph{comonotonicity}. Under comonotonicity and mild additional conditions, we develop a unified enumerative framework showing that fixed-rank convex maximization is polynomially solvable. This viewpoint recovers known results such as convex matroid maximization and also covers mixed-integer applications that previously required separate analyses. For the more structured class of \emph{standard comonotone} sets, we further refine the analysis in a suitably enlarged parameter space and obtain a square-root improvement in the complexity bound. Applications to several PCA-related problems illustrate the broad applicability and effectiveness of the proposed framework.