Improved convergence theorems for bubble clusters: 1. The planar case

We develop an "improved convergence theorem" for a case study variational problem with singularities, namely, the isoperimetric problem on planar bubble clusters. We exploit this theorem in the description of isoperimetric clusters, possibly perturbed by a potential. Our methods are not specific to bubble clusters, and should provide a starting point to address similar issues in other variational problems where minimizers are known to possibly develop singularities. Further applications and extensions are discussed in companion papers.