  We show that a variant of the random-edge pivoting rule results in a strongly
polynomial time simplex algorithm for linear programs $\max\{c^Tx \colon Ax\leq
b\}$, whose constraint matrix $A$ satisfies a geometric property introduced by
Brunsch and R\"oglin: The sine of the angle of a row of $A$ to a hyperplane
spanned by $n-1$ other rows of $A$ is at least $\delta$. This property is a
geometric generalization of $A$ being integral and all sub-determinants of $A$
being bounded by $\Delta$ in absolute value (since $\delta \geq 1/(\Delta^2
n)$). In particular, linear programs defined by totally unimodular matrices are
captured in this famework ($\delta \geq 1/ n$) for which Dyer and Frieze
previously described a strongly polynomial-time randomized algorithm.
  The number of pivots of the simplex algorithm is polynomial in the dimension
and $1/\delta$ and independent of the number of constraints of the linear
program. Our main result can be viewed as an algorithmic realization of the
proof of small diameter for such polytopes by Bonifas et al., using the ideas
of Dyer and Frieze.
