  We show that the graph of a simplicial polytope of dimension $d \ge 3$ has no
nontrivial minimum edge cut with fewer than $d(d+1)/2$ edges, hence the graph
is $\min\{\delta, d(d+1)/2\}$-edge-connected where $\delta$ denotes the minimum
degree. When $d = 3$, this implies that every minimum edge cut in a plane
triangulation is trivial. When $d \ge 4$, we construct a simplicial
$d$-polytope whose graph has a nontrivial minimum edge cut of cardinality
$d(d+1)/2$, proving that the aforementioned result is best possible.
