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We present three results which support the conjecture that a graph isminimally 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 graphbracing operation which preserves independence in the generic rigidity matroidwhen 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 atmost $d+2$ is independent in $\ell_p^d$. Finally, we prove that everytriangulation of the projective plane is minimally rigid in $\ell_p^3$. Acatalogue of rigidity preserving graph moves is also provided for the moregeneral class of strictly convex and smooth normed spaces and we show thatevery triangulation of the sphere is independent for 3-dimensional spaces inthis class.

Which graphs are rigid in $\ell_p^d$?