Let $K$ be a centrally symmetric convex body in $\mathbb R^3$ with volume ${\rm vol}(K)=1$. For any subset $F \subset \lbrace1,2,3\rbrace$, let $K_F$ be the projection of $K$ in $\mathbb R^F$.

Question:What is the best constant $C$, such that$${\rm vol}(K_{\lbrace 1 \rbrace}) \leq C \cdot {\rm vol}(K_{\lbrace 1,2 \rbrace}) \cdot {\rm vol}(K_{\lbrace 1,3 \rbrace}).$$

Here, the volume of $K_F$ is computed in $\mathbb R^F$. With some work one can prove that $C=2$ is good enough. This is elementary and follows for example from two applications of Lemma 3.1 in

J. Bourgain and V.D. Milman, New volume ratio properties for convex symmetric bodies in $\mathbb R^n$, Invent. Math. 1987 vol. 88 (2) pp. 319-340.

My guess is that maybe $C=1$ works, but I am not sure.