Are faces of a compact, convex body "opposed" iff their extreme points are pairwise "opposed"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:19:21Z http://mathoverflow.net/feeds/question/103795 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103795/are-faces-of-a-compact-convex-body-opposed-iff-their-extreme-points-are-pairwi Are faces of a compact, convex body "opposed" iff their extreme points are pairwise "opposed"? Sabri 2012-08-02T16:07:52Z 2012-08-02T21:33:15Z <p>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 <em>opposed</em> if there exist parallel hyperplanes $H_1$, $H_2$ supporting $P$, such that $W_i \subset H_i \cap P$.</p> <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? </p> <p>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).</p> http://mathoverflow.net/questions/103795/are-faces-of-a-compact-convex-body-opposed-iff-their-extreme-points-are-pairwi/103806#103806 Answer by Anton Petrunin for Are faces of a compact, convex body "opposed" iff their extreme points are pairwise "opposed"? Anton Petrunin 2012-08-02T18:22:29Z 2012-08-02T21:33:15Z <p><strong>NO.</strong></p> <p>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 $$F_2=\{\,(0,y,0)\in \mathbb R^3\mid 0\le -y^2+\varepsilon\cdot y+1\,\}$$</p> <p>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.</p>