Rigid graphs in cylindrical normed spaces

We characterise rigid graphs for cylindrical normed spaces $Z=X{\oplus }_{\mathrm{\infty }}\mathbb{R}$ where $X$ is a finite dimensional real normed linear space and $Z$ is endowed with the product norm. In particular, we obtain purely combinatorial characterisations of minimal rigidity for a large class of 3-dimensional cylindrical normed spaces; for example, when $X$ is an ${\mathrm{\ell }}_p$-plane with $p\in (1,\mathrm{\infty })$. We combine these results with recent work of Cros et al. to characterise rigid graphs in the 4-dimensional cylindrical space $({\mathbb{R}}^2{\oplus }_1\mathbb{R}){\oplus }_{\mathrm{\infty }}\mathbb{R}$. These are among the first combinatorial characterisations of rigid graphs in normed spaces of dimension greater than 2. Examples of rigid graphs are presented and algorithmic aspects are discussed.