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