The $n$-Dimensional Cube and the $n$-Dimensional Simplex

Find the shortest distance between a two-dimensional face of the unit four-dimensional cube and its diagonal not intersecting this face.

Prove that the orthogonal projections of the vertices of an $n$-dimensional cube onto any of its diagonals divide it into $n$ equal parts.

Find the angle between the diagonal of a four-dimensional cube and its one-, two-, and three-dimensional faces.

(Pythagorean theorem in $n$-dimensional space). Prove that the square of the diagonal of an $n$-dimensional rectangular parallelepiped equals the sum of the squares of its edges emanating from one vertex.

Find the angle between the diagonal of an $n$-dimensional cube and its $k$-dimensional face.

Find the number of:

Find the length of the diagonal of an $n$-dimensional cube with edge $a$.

What three-dimensional solids are obtained in the cross-section of the four-dimensional cube $\left\{ x \in \mathbb {R}^{4} \mid -1 \leqslant x^{i} \leqslant 1, i = 1,2,3,4\right\}$ by the three-dimensional hyperplane $x^{1}+x^{2}+x^{3}+x^{4} = \varepsilon$ for the values $\varepsilon = 0,2,3,4,5$? Compare the results with the analogous three-dimensional problem.

In the $n$-dimensional cube $I_{1}:\left|x^{i}\right| \leqslant 1, i = 1,2, \ldots , n$, we ``push out'' its walls by $10^{-3}$, i.e., obtain a new cube $I_{1+10^{-3}}:\left|x^{i}\right| \leqslant 1+\varepsilon$, $i = 1,2, \ldots , n$. Find the dimension $n$ at which the volume of the cube $I_{1+10^{-3}}$ is at least twice the volume of the original cube $I_{1}$.

Prove that the line joining the centers of two opposite faces of a regular simplex is perpendicular to these faces.

Find the distance between a $k$-dimensional face of a regular $n$-dimensional simplex with edge of unit length and the opposite $(n-k-1)$-dimensional face.

Find the angle between the two-dimensional faces of a regular four-dimensional simplex.