Just think of the high dimension n-cube like a spiky sea urchin. It has 2^n spikes, and the spheres live in those spikes near the ends. The central sphere is large because it extends out to those spheres, extending outside the sea urchin's "body".
However, it's clear the "starfish" intuition is simply not accurate. That's not what N-cubes look like. The point of this post is that we should have cognitive dissonance when we try to think about 10-cubes, because it's weird that (A) the "inner sphere" pokes out of a shape that is (B) convex everywhere. You can resolve the cognitive dissonance easily by simply ignoring or rejecting B -- sure, it's not weird that such a sphere would poke out of a starfish. But you are wrong. It's not a starfish! It's convex everywhere! So you can't say "why do y'all have cognitive dissonance about this?"
It seems reasonable to extend that same intuition to n-D sharpness/spikiness in an accurate way. Adding an extra “face” just chops more off making those vertices sharper and sharper, at least relative to the high dimensional space around it.
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.
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.