  General relativity's prediction that all black holes are described by the
Kerr metric, irrespective of their size, can now be empirically tested using
electromagnetic observations of supermassive black holes and gravitational
waves from mergers of stellar-mass black holes. In this work, we focus on the
electromagnetic side of this test and quantify the constraining power of
very-long-baseline-interferometry (VLBI) observations of emission generated by
hot gas surrounding supermassive black holes. We demonstrate how to use lensing
bands--annular regions on the observer's screen surrounding the critical
curve--to constrain the underlying spacetime geometry. Contingent upon a
detection of a lensed VLBI feature, the resulting lensing-band framework allows
us to exclude spacetimes for which said feature cannot arise from geodesics
that traversed the equatorial plane more than once. Focusing on the first
indirect image and tests of black-hole uniqueness, we employ a parametrized
spacetime as a case study. We find that resolving geometric information that
goes beyond the apparent size of the critical curve has the potential to lift
degeneracies between different spacetime parameters. Our work thereby
quantifies a conservative estimate of the constraining power of VLBI
measurements and contributes to a larger effort to simultaneously constrain
geometry and astrophysics.
