Estimating heights using auxiliary functions

Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to problems involving the Weil height. The height appears as a solution to a certain extremal problem involving polynomials. We further generalize the above techniques to acquire both the projective height and the height on subspaces in the same way. We further obtain lower bounds on the heights of points on some subvarieties of $\mathbb P^{N-1}(\overline{\mathbb Q})$.