  Let $(x_n)$ be a sequence in a Banach space $X$ which does not converge in
norm, and let $E$ be an isomorphically precisely norming set for $X$ such that
\[ \sum_n |x^*(x_{n+1}-x_n)|< \infty, \; \forall x^* \in E. \qquad (*) \] Then
there exists a subsequence of $(x_n)$ which spans an isomorphically polyhedral
Banach space. It follows immediately from results of V. Fonf that the converse
is also true: If a separable Banach space $Y$ is a separable isomorphically
polyhedral then there exists a non norm convergent sequence $(x_n)$ which spans
$Y$ and there exists an isomorphically precisely norming set $E$ for $Y$ such
that $(*)$ is satisfied. As an application of this subsequence characterization
of sequences spanning isomorphically polyhedral Banach spaces we obtain a
strengthening of a result of J. Elton, and an Orlicz-Pettis type result.
