  Extending on ideas of Lewin, Lieb and Seiringer (Phys Rev B, 100, 035127,
(2019)) we present a modified "floating crystal" trial state for Jellium (also
known as the classical homogeneous electron gas) with density equal to a
characteristic function. This allows us to show that three definitions of the
Jellium energy coincide in dimensions $d\geq 2$, thus extending the result of
Cotar and Petrache (arXiv: 1707.07664) and Lewin, Lieb and Seiringer (Phys Rev
B, 100, 035127, (2019)) that the three definitions coincide in dimension $d
\geq 3$. We show that the Jellium energy is also equivalent to a "renormalized
energy" studied in a series of papers by Serfaty and others and thus, by work
of B\'etermin and Sandier (Constr Approx, 47:39-74, (2018)), we relate the
Jellium energy to the order $n$ term in the logarithmic energy of $n$ points on
the unit 2-sphere. We improve upon known lower bounds for this renormalized
energy. Additionally, we derive formulas for the Jellium energy of periodic
configurations.
