  Light that grazes a black-hole event horizon can loop around one or more
times before escaping again, resulting for distance observers in an infinite
sequence of ever fainter and more delayed images near the black hole shadow. In
the case of the M87 and Sgr A$^*$ back holes, the first of these so-called
photon-ring images have now been observed. A question then arises: are such
images minima, maxima, or saddle-points in the sense of Fermat's principle in
gravitational lensing? or more briefly, the title question above. In the theory
of lensing by weak gravitational fields, image parities are readily found by
considering the time-delay surface (also called the Fermat potential or the
arrival-time surface). In this work, we extend the notion of the time delay
surface to strong gravitational fields and compute the surface for a
Schwarzschild black hole. The time-delay surface is the difference of two
wavefronts, one travelling forward from the source and one travelling backwards
from the observer. Image parities are read off from the topography of the
surface, exactly as in the weak-field regime, but the surface itself is more
complicated. Of the images, furthest from the black hole and similar to the
weak-field limit, are a minimum and a saddle point. The strong field repeats
the pattern, corresponding to light taking one or more loops around the back
hole. In between, there are steeply-rising walls in the time-delay surface,
which can be interpreted as maxima and saddle points that are infinitely
delayed and not observable -- these correspond to light rays taking a U-turn
around the black hole.
