  This dissertation investigates the geometric combinatorics of convex
polytopes and connections to the behavior of the simplex method for linear
programming. We focus our attention on transportation polytopes, which are sets
of all tables of non-negative real numbers satisfying certain summation
conditions. Transportation problems are, in many ways, the simplest kind of
linear programs and thus have a rich combinatorial structure. First, we give
new results on the diameters of certain classes of transportation polytopes and
their relation to the Hirsch Conjecture, which asserts that the diameter of
every $d$-dimensional convex polytope with $n$ facets is bounded above by
$n-d$. In particular, we prove a new quadratic upper bound on the diameter of
$3$-way axial transportation polytopes defined by $1$-marginals. We also show
that the Hirsch Conjecture holds for $p \times 2$ classical transportation
polytopes, but that there are infinitely-many Hirsch-sharp classical
transportation polytopes. Second, we present new results on subpolytopes of
transportation polytopes. We investigate, for example, a non-regular
triangulation of a subpolytope of the fourth Birkhoff polytope $B_4$. This
implies the existence of non-regular triangulations of all Birkhoff polytopes
$B_n$ for $n \geq 4$. We also study certain classes of network flow polytopes
and prove new linear upper bounds for their diameters.
