Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W\_1$, $W\_2$ $\subset P$ are *opposed* if there exist parallel hyperplanes $H\_1$, $H\_2$ supporting $P$, such that $W\_i \subset H\_i \cap P$.


Let $F\_1$ and $F\_2$ be faces of $P$, such that their extreme points are pairwise opposed: i.e. $v\_1, v\_2 \in P$ are opposed whenever $v\_i$ is an extreme point of $F\_i$. Are $F\_1$ and $F\_2$ opposed? 


I have a tentative proof when $P$ has affine dimension equal to 2, which I am struggling to generalise even to 3 dimensions. The converse is trivial. I'd also be interested to know if a proof of this requires some restriction on $P$ (e.g. letting $P$ be a polytope).



Are faces of a compact, convex body "opposed" iff their extreme points are pairwise "opposed"?
**NO.**


Let $P$ be the convex hull of two parabolic arcs,
say 
$$\{\,(x,0,z)\in \mathbb R^3\mid 1\ge z=x^2\,\}$$
and 
$$\{\,(0,y,z)\in \mathbb R^3\mid 0\le z=-y^2+\varepsilon\cdot y+1\,\}.$$
Take
$$F\_1=\{\,(x,0,1)\in \mathbb R^3\mid |x|\le 1\,\}$$
and 
$$F\_2=\{\,(0,y,0)\in \mathbb R^3\mid 0\le -y^2+\varepsilon\cdot y+1\,\}$$


You can approximate it by a polyhedra in such a way that $F\_1$ and $F\_2$ are still edges.
This leads to a polyhedral example.


