We quantify the shape and overdensity of the galaxy distribution in the `Local Sheet within a sphere of $R=8$ Mpc, and compare these properties with the expectations of the $Lambda$CDM model. We measure ellipsoidal axis ratios of $c/aapprox0.16$ and $b/aapprox0.79$, indicating that the distribution of galaxies in the Local Volume can be approximated by a flattened oblate ellipsoid, consistent with the `sheet-like configuration noted in previous studies. In contrast with previous estimates that the Local Sheet has a density close to average, we find that the number density of faint and bright galaxies in the Local Volume is $approx1.7$ and $approx5.2$ times denser, respectively, than the mean number density of galaxies of the same luminosity. Comparison with simulations shows that the number density contrasts of bright and faint galaxies within $8$ Mpc alone make the Local Volume a $approx 2.5sigma$ outlier in the $Lambda$CDM cosmology. Our results indicate that the cosmic neighbourhood of the Milky Way may be unusual for galaxies of similar luminosity. The impact of the peculiar properties of our neighbourhood on the properties of the Milky Way and other nearby galaxies is not yet understood and warrants further study.
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