  There is an exciting prospect of obtaining the shadow of astrophysical black
holes (BHs) in the near future with the Event Horizon Telescope. As a matter of
principle, this justifies asking how much one can learn about the BH horizon
itself from such a measurement. Since the shadow is determined by a set of
special photon orbits, rather than horizon properties, it is possible that
different horizon geometries yield similar shadows. One may then ask how
sensitive is the shadow to details of the horizon geometry? As a case study, we
consider the double Schwarzschild BH and analyse the impact on the lensing and
shadows of the conical singularity that holds the two BHs in equilibrium --
herein taken to be a strut along the symmetry axis in between the two BHs.
Whereas the conical singularity induces a discontinuity of the scattering angle
of photons, clearly visible in the lensing patterns along the direction of the
strut's location, it produces no observable effect on the shadows, whose edges
remain everywhere smooth. The latter feature is illustrated by examples
including both equal and unequal mass BHs. This smoothness contrasts with the
intrinsic geometry of the (spatial sections of the) horizon of these BHs, which
is not smooth, and provides a sharp example on how BH shadows are insensitive
to some horizon geometry details. This observation, moreover, suggests that for
the study of their shadows, this static double BH system may be an informative
proxy for a dynamical binary.
