  A well-known inner bound on the stability region of the finite-user slotted
Aloha protocol is the set of all arrival rates for which there exists some
choice of the contention probabilities such that the associated worst-case
service rate for each user exceeds the user's arrival rate, denoted $\Lambda$.
Although testing membership in $\Lambda$ of a given arrival rate can be posed
as a convex program, it is nonetheless of interest to understand the properties
of this set. In this paper we develop new results of this nature, including
$i)$ an equivalence between membership in $\Lambda$ and the existence of a
positive root of a given polynomial, $ii)$ a method to construct a vector of
contention probabilities to stabilize any stabilizable arrival rate vector,
$iii)$ the volume of $\Lambda$, $iv)$ explicit polyhedral, spherical, and
ellipsoid inner and outer bounds on $\Lambda$, and $v)$ characterization of the
generalized convexity properties of a natural ``excess rate'' function
associated with $\Lambda$, including the convexity of the set of contention
probabilities that stabilize a given arrival rate vector.
