Faces, diagonals and cross-sections of cubes and parallelepipeds

Prove the convexity of a $k$-dimensional parallelepiped.

Find the projection of the four-dimensional simplex bounded by the hyperplanes $x_{1} = 0, x_{2} = 0, x_{3} = 0, x_{4} = 0$ and $x_{1} + x_{2} + x_{3} + x_{4} = 1$, onto the hyperplane $x_{1} + x_{2} + x_{3} + x_{4} = 0$ parallel to the line $x_{1} = x_{2} = x_{3} = x_{4}$.

Prove that all diagonals of a parallelepiped intersect at one point, called the center of the parallelepiped.

For a $k$-dimensional parallelepiped, find the number of:

Determine the shape and vertices of the cross-sections of the four-dimensional parallelepiped $- 1 \leqslant x_{i} \leqslant 1, i = 1, 2, 3, 4$, by the hyperplane $x_{1} + x_{2} + x_{3} + x_{4} = 0$.

Let $a_{0}, a_{1}, \ldots , a_{n}$ be points of an $n$-dimensional affine space in general position, $H_{i}(i=0,1, \ldots , n)$ the hyperplane passing through all these points except $a_{i}$, and $H_{i}^{+}$ the half-space bounded by it that contains the point $a_{i}$. Prove that

Find the faces of an $n$-dimensional parallelepiped given, in some system of affine coordinates, by the inequalities $0 \leqslant x_{i} \leqslant 1(i=1$, $2, \ldots , n)$.

Find the vertices and describe the shape of the sections of the four-dimensional parallelepiped given by the inequalities $0 \leqslant x_{i} \leqslant 1(i=1,2,3,4)$, by the hyperplanes:

Find the number of:

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 $I_{1}$.