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Register a new userA note on the extremal noncentral sections of the cross-polytope
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We find minimal and maximal length of intersections of lines at a fixed distance to the origin with the cross-polytope. We also find maximal volume noncentral sections of the cross-polypote by hyperplanes which are at a fixed large distance to the origin and minimal volume sections by symmetric slabs of a large fixed width. This parallels recent results about noncentral sections of the cube due to Moody, Stone, Zach and Zvavitch.
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The no-signaling polytope associated to a Bell scenario with three parties, two inputs, and two outputs is found to have 53856 extremal points, belonging to 46 inequivalent classes. We provide a classification of these points according to various definitions of multipartite non-locality and briefly discuss other issues like the interconversion between extremal points seen as a resource and the relation of the extremal points to Bell-type inequalities.
Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well known that for any convex polygon there exist at least two empty and at least two full circles, i.e. at least four extremal circles. In 1990 Schatteman considered a generalization of this theorem for convex polytopes in d-dimensional Euclidean space. Namely, he claimed that there exist at least 2d extremal neighboring spheres. In this paper, we show that there are certain gaps in Schattemans proof, which is based on the Bruggesser-Mani shelling method. We show that using this method it is possible to prove that there are at least d+1 extremal neighboring spheres. However, the existence problem of 2d extremal neighboring spheres is still open.
Let $X$ be a geodesic metric space with $H_1(X)$ uniformly generated. If $X$ has asymptotic dimension one then $X$ is quasi-isometric to an unbounded tree. As a corollary, we show that the asymptotic dimension of the curve graph of a compact, oriented surface with genus $g ge 2$ and one boundary component is at least two.
Our starting point is an improved version of a result of D. Hajela related to a question of Koml{o}s: we show that if $f(n)$ is a function such that $limlimits_{ntoinfty }f(n)=infty $ and $f(n)=o(n)$, there exists $n_0=n_0(f)$ such that for every $ngeqslant n_0$ and any $Ssubseteq {-1,1}^n$ with cardinality $|S|leqslant 2^{n/f(n)}$ one can find orthonormal vectors $x_1,ldots ,x_nin {mathbb R}^n$ that satisfy $$|epsilon_1x_1+cdots +epsilon_nx_n|_{infty }geqslant csqrt{log f(n)}$$ for all $(epsilon_1,ldots ,epsilon_n)in S$. We obtain analogous results in the case where $x_1,ldots ,x_n$ are independent random points uniformly distributed in the Euclidean unit ball $B_2^n$ or any symmetric convex body, and the $ell_{infty }^n$-norm is replaced by an arbitrary norm on ${mathbb R}^n$.
The first author showed that for a given point $p$ in an $nk$-polytope $P$ there are $n$ points in the $k$-faces of $P$, whose barycenter is $p$. We show that we can increase the dimension of $P$ by $r$, if we allow $r$ of the points to be in $(k+1)$-faces. While we can force points with a prescribed barycenter into faces of dimensions $k$ and $k+1$, we show that the gap in dimensions of these faces can never exceed one. We also investigate the weighted analogue of this question, where a convex combination with predetermined coefficients of $n$ points in $k$-faces of an $nk$-polytope is supposed to equal a given target point. While weights that are not all equal may be prescribed for certain values of $n$ and $k$, any coefficient vector that yields a point different from the barycenter cannot be prescribed for fixed $n$ and sufficiently large $k$.
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