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Motivated by the observation that groups can be effectively studied usingmetric spaces modelled on $\ell^1$, $\ell^2$, and $\ell^\infty$ geometry, weconsider cell complexes equipped with an $\ell^p$ metric for arbitrary $p$.Under weak conditions that can be checked locally, we establish nonpositivecurvature properties of these complexes, such as Busemann-convexity and strongbolicity. We also provide detailed information on the geodesics of thesemetrics in the special case of CAT(0) cube complexes.

$\ell^p$ metrics on cell complexes