Where the volume of a high-dimensional cube lives

Flat projections of the n-dimensional cube (dots are corners, lines are edges). The green band is the outer layer holding the chosen share of the volume. As n grows, the band gets thinner and hugs the boundary.

The cube [-1,1]^n has inner sub-cube [-a,a]^n holding a^n of its volume, so a band with share s needs a = (1-s)^(1/n). Its thickness 1-a shrinks roughly like 2.3/n for 90%. The drawing is an orthogonal projection, so the inner outline is the exact shadow of the sub-cube; the band is what lies between the outlines, a sketch of the boundary layer and not a literal shadow of it.