Subspace of $\mathbb{R}^n$ spanned by the image of convex $(n-1)$-polyhedra under the face-counting map - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:35:23Z http://mathoverflow.net/feeds/question/58688 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58688/subspace-of-mathbbrn-spanned-by-the-image-of-convex-n-1-polyhedra-under Subspace of $\mathbb{R}^n$ spanned by the image of convex $(n-1)$-polyhedra under the face-counting map The Cheese Stands Alone 2011-03-16T21:35:24Z 2011-03-17T14:12:46Z <p>Fix $n \in \mathbb{N}$. A convex polyhedron $C$ in $\mathbb{R}^n$ is the convex hull of finitely many points with nonempty interior. For $H$ a supporting hyperplane, ie $C$ is contained in one of the two closed half-spaces bounded by $H$, we call $H \cap C$ a $j$-face of $C$, where $j$ is the affine dimension of $H \cap C$. By convention, $\varnothing$ is called a $-1$-face of $C$ and $C$ an $n$-face of itself.</p> <p>Define a function $F$ from the set of convex polyhedra to $\mathbb{R}^{n+2}$ by coordinates, so that $F(C) = (a^C_{-1}, ..., a^C_n)$, where $a^C_j$ is the number of $j$-faces of $C$ for $j=-1,...,n$. Let $W$ be the affine subspace of $\mathbb{R}^{n+2}$ generated by $\operatorname{im} F$.</p> <p>It's clear that $a^C_{-1}=1$ and $a^C_n=1$. Euler's formula $\displaystyle \sum_{j=-1}^n (-1)^j a^C_j = 0$ (which may be more familiar as the Euler characteristic $V+E-F=2$ in the case of $n=3$) is a third affine relation between the $a^C_j$'s. Hence, $\operatorname{dim}W \le n-1$. </p> <p>Is it always true for any n that $\operatorname{dim}W = n-1$? Put differently, for any $n$, are the three equations above the only affine relationships that must be satisfied by $a^C_j$'s for all convex polyhedra $C \subset \mathbb{R}^n$, or is there some $n$ in which there is another relation?</p> <p>I seem to recall an affirmative answer to this, but I can't remember how it was solved or where I found it.</p> http://mathoverflow.net/questions/58688/subspace-of-mathbbrn-spanned-by-the-image-of-convex-n-1-polyhedra-under/58731#58731 Answer by Gjergji Zaimi for Subspace of $\mathbb{R}^n$ spanned by the image of convex $(n-1)$-polyhedra under the face-counting map Gjergji Zaimi 2011-03-17T08:38:23Z 2011-03-17T14:12:46Z <p>Yes this is always true. One just needs to exhibit enough polyhedra so that the span of their f-vectors is $n-1$ dimensional. Such a family is given by the polyhedra $\Delta^k\times I^{n-k}$, where $\Delta^k$ is the $k$-simplex and $I^{k}$ is the $k$-cube.</p> <p>The same argument can be used that the corresponding dimension for simplicial polyhedra is $\lfloor \frac{n}{2}\rfloor +1$, and so the only affine relations are the <a href="http://en.wikipedia.org/wiki/Dehn-Sommerville_equations" rel="nofollow">Dehn-Sommerville relations</a>. One looks at the family $\Delta^k\times \Delta^{n-k}$, $k=0,1,\dots,\lfloor\frac{n}{2}\rfloor$.</p>