It is needed to prove that for set $A \subset N^4$ $3\log|A|\le \log|\pi_{1,2,3}(A)|+\log|\pi_{1,2,4}(A)|+\log|\pi_{1,3,4}(A)|+\log|\pi_{2,3,4}(A)|$ where $\pi_{i,j,k}(A)$ is a projection of $A$ on coordinate plane $i,j,k$ and |X| means the cardinality of the set X.
I think that I found the decision, but I have some doubts about its correctness.
My decision is:
\begin{align} & |A| \le |\pi_{i,j,k}(A)| \cdot |\pi_{l}(A)| \\[8pt] \Longrightarrow \quad |A|^3 & \le |\pi_{1,2,3}(A)| |\pi_4(A)| \cdot |\pi_{2,3,4}(A)| |\pi_1(A)| \cdot |\pi_{1,2,4}(A)| |\pi_{3}(A)| \\[8pt] & \le |\pi_{1,2,3}(A)| \cdot |\pi_{2,3,4}(A)| \cdot |\pi_{1,2,4}(A)| \cdot |\pi_{1,3,4}(A)| \tag{$*$} \end{align} Here $\pi_i(A)$ is projection of A on i axis and ($\cdot$) is a sign of direct product. Thus $3\log|A| \le \log|\pi_{1,2,3}(A)| + \log|\pi_{1,2,4}(A)| + \log|\pi_{2,3,4}(A)| + log|\pi_{1,3,4}(A)$.
My doubts are linked with the last transition in inequalities $(*)$. It is possible if $|\pi_1(A)| \cdot |\pi_3(A)| \cdot |\pi_4(A)| \le |\pi_{1,3,4}(A)|$ or $|\pi_1(A)| \cdot |\pi_3(A)| \cdot |\pi_4(A)| = |\pi_{1,3,4}(A)|$ but I am not sure that it is true.
May be it will be helpful if I told that it is linked with a course of information-theory where logarithm from a cardinality of a set is considered as Hartley information.
Also this question may be changed into a question about proving that volume of any 4-dimensional shape in third degree is equal to a product of squares of projections of this shape on four 3-dimensional planes. Theese planes are defined by equalities i=0, j=0, k=0, l=0 where i, j, k, l are axis koordinta.