  Using the Fenchel-Eggleston theorem for convex hulls (an extension of the
Caratheodory theorem), we prove that any likelihood can be maximized by either
a dark matter 1- speed distribution $F(v)$ in Earth's frame or 2- Galactic
velocity distribution $f^{\rm gal}(\vec{u})$, consisting of a sum of delta
functions. The former case applies only to time-averaged rate measurements and
the maximum number of delta functions is $({\mathcal N}-1)$, where ${\mathcal
N}$ is the total number of data entries. The second case applies to any
harmonic expansion coefficient of the time-dependent rate and the maximum
number of terms is ${\mathcal N}$. Using time-averaged rates, the
aforementioned form of $F(v)$ results in a piecewise constant unmodulated halo
function $\tilde\eta^0_{BF}(v_{\rm min})$ (which is an integral of the speed
distribution) with at most $({\mathcal N}-1)$ downward steps. The authors had
previously proven this result for likelihoods comprised of at least one
extended likelihood, and found the best-fit halo function to be unique. This
uniqueness, however, cannot be guaranteed in the more general analysis applied
to arbitrary likelihoods. Thus we introduce a method for determining whether
there exists a unique best-fit halo function, and provide a procedure for
constructing either a pointwise confidence band, if the best-fit halo function
is unique, or a degeneracy band, if it is not. Using measurements of modulation
amplitudes, the aforementioned form of $f^{\rm gal}(\vec{u})$, which is a sum
of Galactic streams, yields a periodic time-dependent halo function
$\tilde\eta_{BF}(v_{\rm min}, t)$ which at any fixed time is a piecewise
constant function of $v_{\rm min}$ with at most ${\mathcal N}$ downward steps.
In this case, we explain how to construct pointwise confidence and degeneracy
bands from the time-averaged halo function. Finally, we show that requiring an
isotropic ...
