Sections and convex hulls of convex bodies

Let $I^N$ be the $N$-dimensional cube defined in $\mathbb {R}^{N}$ by the inequalities $\left|x_i\right|\leq 1$, $1\leq i\leq N$. Prove that any convex bounded subset of the plane can be approximated to any degree of accuracy by two-dimensional sections of $I^N$. (More precisely, for any $\varepsilon >0$ and any convex set $V\subset \mathbb {R}^{2}$ there exist an $N$ and an imbedding $\varphi :\mathbb {R}^{2}\to \mathbb {R}^{N}$ such that the Minkowski functionals of the sets $U$ and $V=\varphi^{-1}(I^N)$ are connected by the inequalities $1-\varepsilon <p_U/p_V<1+\varepsilon$.)

Let $A \subset \mathbb {R}^{m}, B \subset \mathbb {R}^{n}$ be convex compact centrally symmetric sets. What fraction of the volume of the set $A \times B$ is the volume of the convex hull of the set $(A \times \left\{ 0\right\} ) \cup (\left\{ 0\right\} \times B)$?