We report a molecular dynamics (MD) simulation study of protein adsorption on the surface of nanosized carbon allotropes, namely single-walled carbon nanotubes (SWNT) considering both the convex outer surface and the concave inner surface, together with a graphene sheet for comparison. These systems are chosen to investigate the effect of the surface curvature on protein adsorption at the same surface chemistry, given by sp2 carbon atoms in all cases. The simulations show that proteins do favorably interact with these hydrophobic surfaces, as previously found on graphite which has the same chemical nature. However, the main finding of the present study is that the adsorption strength does depend on the surface topography: in particular, it is slightly weaker on the outer convex surfaces of SWNT and is conversely enhanced on the inner concave SWNT surface, being therefore intermediate for flat graphene. We additionally find that oligopeptides may enter the cavity of common SWNT, provided their size is small enough and the tube diameter is large enough for both entropic and energetic reasons. Therefore, we suggest that proteins can effectively be used to solubilize in water single-walled (and by analogy also multiwalled) carbon nanotubes through adsorption on the outer surface, as indeed experimentally found, and to functionalize them after insertion of oligopeptides within the cavity of nanotubes of appropriate size.