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.
> It depends on how you define "concaveness". A cube is concaver than a square in a sense.
Yup. Even a regular 3D cube (and 2D square) has concave faces if you're viewing it from a polar perspective. As I stand in the center of the cube, measuring distances from me to the surface, I'll see that measurement follow a concave pattern.
Yeah, I know that's not the definition of concavity or whatever, but when relating a sphere to a cube, and trying to get an intuition of higher-dimensional spaces, I think it helps to look at it from the sphere's perspective rather than a Cartesian one.
Thank you! This comment solidified for me that the increasing ratio of distances from origin to vertex/side together with the fact that the hypercube surface is still not curved in any way whatsoever is the major thing my spatial intuition (unsurprisingly) struggles with.
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.