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.

我们提出了三个结果，支持图在d维\(\ell _p\)空间中具有最小刚性的猜想，其中\(p\in (1,\infty )\)和\(p\not =2 \) ，当且仅当它是 ( d , d ) 紧的。首先，我们引入一种图支撑操作，当从\(\ell _p^d\)传递到\(\ell _p^{d+1}\)时，该操作在通用刚性拟阵中保持独立性。然后我们证明每个最小度最多为\(d+1\)且最大度最多为\(d+2\) 的( d , d ) 稀疏图在\(\ell _p^d\)中是独立的。最后，我们证明射影平面的每个三角剖分在\(\ell _p^3\)中都是最小刚性的。还为更一般的严格凸和光滑赋范空间类提供了保持刚度的图移动目录，并且我们表明球体的每个三角剖分对于此类中的 3 维空间都是独立的。