On the Whitney extension problem for near isometries and beyond

In this memoir, we develop a general framework which allows for a simultaneous study of labeled and unlabeled near alignment data problems in $\mathbb R^D$ and the Whitney near isometry extension problem for discrete and non-discrete subsets of $\mathbb R^D$ with certain geometries. In addition, we survey related work of ours on clustering, dimension reduction, manifold learning, vision as well as minimal energy partitions, discrepancy and min-max optimization. Numerous open problems in harmonic analysis, computer vision, manifold learning and signal processing connected to our work are given. A significant portion of the work in this memoir is based on joint research with Charles Fefferman in the papers [48], [49], [50], [51].