Revisiting the minimum-norm problem
The design of optimal Magnetic Resonance Imaging (MRI) coils is modeled as a minimum-norm problem (MNP), that is, as an optimization problem of the form min(x is an element of R) parallel to x parallel to, where R is a closed and convex subset of a normed space X. This manuscript is aimed at revisiting MNPs from the perspective of Functional Analysis, Operator Theory, and Banach Space Geometry in order to provide an analytic solution to the following MRI problem: min(psi is an element of R) parallel to psi parallel to(2), where R:= {psi is an element of R-n : parallel to A psi-b parallel to infinity/parallel to b parallel to(infinity) 0, and b is an element of R-m \ {0}. ; This work has been supported by the Research Grant PGC-101514-B-I00 awarded by the Spanish Ministry of Science, Innovation and Universities and partially funded by ERDF, by the Andalusian Research, Development and Innovation Programme (PAIDI 2020) under the Research Grant PY20_01295, and by the Research Grant FEDER-UCA18-105867 awarded by the 2014-2020 ERDF Operational Programme of the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia. The APCs have been paid by the Mathematics Department of the University of Cadiz.