A compact subset of $\mathbb{C}^n$ is polynomially convex if it is defined by a (possibly infinite) family of polynomial inequalities. In this talk, we will discuss some questions and recent results regarding the minimum embedding (complex) dimension of abstract compact (real) manifolds subject to such convexity constraints. These embeddings arise naturally in connection with higher-dimensional analogues of the fact that the algebra of continuous functions on a circle can be generated by two smooth functions. We will expand on this connection, discuss the local and global challenges in constructing such embeddings, and discuss two distinct cases in which new bounds have been obtained. This is joint work with R. Shafikov.