  We consider a generalization of the concept of $d$-flattenability of graphs -
introduced for the $l_2$ norm by Belk and Connelly - to general $l_p$ norms,
with integer $P$, $1 \le p < \infty$, though many of our results work for
$l_\infty$ as well. The following results are shown for graphs $G$, using
notions of genericity, rigidity, and generic $d$-dimensional rigidity matroid
introduced by Kitson for frameworks in general $l_p$ norms, as well as the
cones of vectors of pairwise $l_p^p$ distances of a finite point configuration
in $d$-dimensional, $l_p$ space: (i) $d$-flattenability of a graph $G$ is
equivalent to the convexity of $d$-dimensional, inherent Cayley configurations
spaces for $G$, a concept introduced by the first author; (ii)
$d$-flattenability and convexity of Cayley configuration spaces over specified
non-edges of a $d$-dimensional framework are not generic properties of
frameworks (in arbitrary dimension); (iii) $d$-flattenability of $G$ is
equivalent to all of $G$'s generic frameworks being $d$-flattenable; (iv)
existence of one generic $d$-flattenable framework for $G$ is equivalent to the
independence of the edges of $G$, a generic property of frameworks; (v) the
rank of $G$ equals the dimension of the projection of the $d$-dimensional
stratum of the $l_p^p$ distance cone. We give stronger results for specific
norms for $d = 2$: we show that (vi) 2-flattenable graphs for the $l_1$-norm
(and $l_\infty$-norm) are a larger class than 2-flattenable graphs for
Euclidean $l_2$-norm case and finally (vii) prove further results towards
characterizing 2-flattenability in the $l_1$-norm. A number of conjectures and
open problems are posed.
