  Of the various energy conditions which can be assumed when studying
mathematical general relativity, intuitively the simplest is the weak energy
condition $\mu\geq 0$ which simply states that the observed mass-energy density
must be non-negative. This energy condition has not received as much attention
as the so-called dominant energy condition. When the natural question of the
Penrose inequality in the context of the weak energy condition arose, we could
not find any results in the literature, and it was not immediately clear
whether the inequality would hold, even in spherical symmetry. This led us to
constructing a spherically symmetric asymptotically flat initial data set
satisfying the weak energy condition which violates the "usual" formulations of
the Penrose conjecture. The construction itself is quite elementary. However,
the consequences of the counterexample are quite interesting. The Penrose
inequality was conjectured by Penrose in \cite{Penrose} using certain heuristic
arguments which, as we discuss, continue to hold in the case of the weak energy
condition. Yet, a counter example exists. Moreover, the methods developed
naturally led to the construction of a counter example to the positive mass
theorem assuming the weak energy condition. The counter example can be
constructed to be diffeomorphic to $\mathbb{R}^3$ and to contain no minimal
surfaces and no future apparent horizons.
