Abstract: In 1996 Laszlo Erdoes showed that among planar domains of fixed area, the smallest principal eigenvalue of the Dirichlet Laplacian with a constant magnetic field is uniquely achieved on the disk (Calc. Var. Partial Differ. Equ. 4, 283-292). I now present a quantitative version of his inequality, with an explicit remainder term depending on the field strength that measures how much the domain deviates from being a disk. This tells us, in particular, that if the principal eigenvalue of the Dirichlet magnetic Laplacian is just slightly larger on a planar domain than on the disk of same area--and the field strength is not too large--then that domain is only slightly different from the disk. Faint perturbations of the smallest principal eigenvalue will not induce a dramatic change in the underlying geometry. The talk is based on joint work with Lukas Junge and Leo Morin (Aarhus).