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arXiv · 2402.18190

Generic Global Rigidity in $\ell_p$-Space and the Identifiability of the $p$-Cayley-Menger Varieties

Tomohiro Sugiyama·Shin-ichi Tanigawa
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Abstract

The celebrated result of Gortler-Healy-Thurston (independently, Jackson-Jord\'an for $d=2$) shows that the global rigidity of graphs realised in the $d$-dimensional Euclidean space is a generic property. Extending this result to the global rigidity problem in $\ell_p$-spaces remains an open problem. In this paper we affirmatively solve this problem when $d=2$ and $p$ is an even positive integer. A key tool in our proof is a sufficient condition for the $d$-tangential weak non-defectivity of projective varieties due to Bocci, Chiantini, Ottaviani, and Vannieuwenhoven. By specialising the condition to the $p$-Cayley-Menger variety, which is the $\ell_p$-analogue of the Cayley-Menger variety for Euclidean distance, we provide an $\ell_p$-extension of the generic global rigidity theory of Connelly. As a by-product of our proof, we also offer a purely graph-theoretical characterisation of the $2$-identifiability of an orthogonal projection of the $p$-Cayley-Menger variety along a coordinate axis of the ambient affine space.

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Tomohiro Sugiyama, Shin-ichi Tanigawa. 2024-02-28. Generic Global Rigidity in $\ell_p$-Space and the Identifiability of the $p$-Cayley-Menger Varieties. https://arxiv.org/abs/2402.18190

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