  We present three results which support the conjecture that a graph is
minimally rigid in $d$-dimensional $\ell_p$-space, where $p\in (1,\infty)$ and
$p\not=2$, if and only if it is $(d,d)$-tight. Firstly, we introduce a graph
bracing operation which preserves independence in the generic rigidity matroid
when passing from $\ell_p^d$ to $\ell_p^{d+1}$. We then prove that every
$(d,d)$-sparse graph with minimum degree at most $d+1$ and maximum degree at
most $d+2$ is independent in $\ell_p^d$. Finally, we prove that every
triangulation of the projective plane is minimally rigid in $\ell_p^3$. A
catalogue of rigidity preserving graph moves is also provided for the more
general class of strictly convex and smooth normed spaces and we show that
every triangulation of the sphere is independent for 3-dimensional spaces in
this class.
