[sillymath]

> Like, you can inscribe a circle inside a square and have it touch all sides without extending outside the square, and you can similarly “inscribe” a sphere inside a cube such that the sphere touches all sides of the cube without extending outside the cube. Does this hold true in higher dimensions?

Of course, points with only one nonzero coordinates xi in $\{1,-1}$ are in both the sphere and the sides of the cube, and those are the minimal distance points from the sides of the cube to the center of the circle (since they are orthogonal to the hyperplane xi=+1, or xi=-1 that contains it). That also shows that no side of the cube extends outside of the circle. Some more math: A well known result in math is that the minimum distance from the origin O to a linear subspace of R^n is attained at points P such that OP is orthogonal to the direction of the subspace. In this case, since the linear space is a hyperplane there is just one orthogonal direction to that hyperplane, and there is just one point in the intersection of the hyperplane and the line defined by the point O and the orthogonal direction to the hyperplane, and that intersection is just the points we alluded before.