Unified halo-independent formalism from convex hulls for direct dark matter searches

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…