\section{Conclusion}
After providing a short review of generalized convexity and its applications, we developed a framework for parameterizing generalized convex functions. We showed that finitely $Y$-convex functions form a dense subset of all $Y$-convex functions, establishing their universal approximation property for both generalized convex functions and their gradients under mild regularity conditions. We also demonstrated that the parameterization has a convex parameter space and can be made injective.

We highlighted structural parallels between finitely convex parameterizations and shallow neural networks, suggesting that deeper analogues may offer practical advantages, as observed with deep networks in machine learning.

On the applied side, the methods presented were implemented in the \href{https://github.com/MoeenNehzati/gconvex}{\texttt{gconvex}} package and used for finding revenue-maximizing mechanisms in multi-item auctions. We compared the results with existing benchmarks and found that our approach achieves virtually the same profits.

This work provides a foundation for future applied research in mathematical economics and optimal transport. Finding deep architectures for finitely convex functions and answering whether, similar to neural networks, local methods find global optima are important questions left to future work.