It depends on how you define "concaveness". A cube is concaver than a square in a sense. The travel from the center of a unit square to its side takes 0.5, and to its vertex ~0.7. For a cube, 0.5 and ~0.87. For a 100d cube, 0.5 and 5. (For completeness, for 1d cube it's 0.5, 0.5). Of course that is not a true concaveness, but it gives a nice sense of inside distances, especially with spheres, which are defined as equidistant.

We are just used to project a cube in a way that prevents to see its linearly-spatial configuration (a projection messes up lengths), but if you preserve these lengths and "flatten" them instead, a cube will flatten out to a sort of a shuriken.