We study the double discrete Legendre--Fenchel Transform (LFT) to approximate the convex hull of a given function. We analyze the convergence of the double discrete LFT in the multivariate case based on previous convergence results for the discrete LFT by Corrias. We focus our attention on the grid on which the second discrete LFT is computed (dual grid); its choice has great impact on the accuracy of the resulting approximation of the convex hull. Then, we present an improvement (both in time and accuracy) to the standard algorithm based on a change in the factorization order for the second discrete LFT. This modification is particularly beneficial for bivariate functions. We also present some situations in which the selection of the dual grid is crucial, and show that it is possible to choose a dual grid of arbitrary size without increasing the memory requirements of the algorithm. Finally, we apply our algorithm to the study of phase separation in ionic solutions where non-ideal effects due to long-range electrostatic and short-range steric correlations between ions play an important role.