  Our motivating goal is factorization in Krull Domains $H$ with finitely
generated class group $G$. The elasticity $\rho(H)$ is the maximal number of
atoms in any re-factorization of a product of $k$ atoms. The elasticities are
the same as those of a combinatorial monoid of zero-sum sequences $B(G_0)$,
where $G_0\subseteq G$ are the classes with height one primes. We characterize
when finite elasticity holds for any Krull Domain with finitely generated class
group. Our results are valid for the more general class of Transfer Krull
Monoids (over a subset $G_0$ of a finitely generated abelian group $G$). We
show there is a minimal $s\leq (d+1)m$, where $d$ is the torsion free rank and
$m$ is the torsion exponent, such that $\rho_s(H)<\infty$ implies
$\rho_k(H)<\infty$ for all $k\geq 1$. This ensures $\rho(H)<\infty$ if and only
if $\rho_{(d+1)m}(H)<\infty$. Our characterization is in terms of a simple
combinatorial obstruction to infinite elasticity: there existing a subset
$G_0^\diamond\subseteq G_0$ and bound $N$ such that there are no nontrivial
zero-sum sequences with terms from $G_0^\diamond$, and every minimal zero-sum
sequence has at most $N$ terms from $G_0\setminus G_0^\diamond$. We give an
explicit description of $G_0^\diamond$ in terms of the Convex Geometry of $G_0$
modulo the torsion subgroup $G_T\leq G$, and show finite elasticity is
equivalent to there being no positive linear combination of the elements of
this explicitly defined subset equal to $0$ modulo $G_T$. We use our results to
show finite elasticity implies the set of distances $\Delta(H)$, the catenary
degree $\mathsf c(H)$ (for Krull Monoids) and a weakened tame degree (for Krull
Monoids) are all also finite, and that the Structure Theorem for Unions holds.
Our results for factorization in Transfer Krull Monoids are accomplished by
developing an extensive theory in Convex Geometry generalizing positive bases.
