  In the past decade, much effort has gone into understanding maximizers for
Fourier restriction and extension inequalities. Nearly all of the cases in
which maximizers for inequalities involving the restriction or extension
operator have been successfully identified can be seen as partial answers to
the question in the title. In this survey, we focus on recent developments in
sharp restriction theory relevant to this question. We present results in the
algebraic case for spherical and hyperbolic extension inequalities. We also
discuss the use of the Penrose transform leading to some negative answers in
the case of the cone.
