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        Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.

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        24 views

        Defining a notion of “volume of its lattice” for non-rational subspaces

        Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”: $$\...
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        1answer
        128 views

        Random reflections unexpectedly produce banded distributions

        Start with $p_1$ a random point on the origin-centered unit circle $C$. At step $i$, select a random point $q_i$ on $C$, and a random mirror line $M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...
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        54 views

        On Topological Tverberg Theorem

        Let $r$ be a prime power. The Topological Tverberg Theorem says that any continuous map from a $(r-1)(d+1)$-simplex to $\mathbb{R}^d$ identifies points from $r$ disjoint faces. It is not hard to see ...
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        101 views

        Does the problem of recognizing 3DORG-graphs have polynomial complexity?

        A 2DORG is the intersection graph of a finite family of rays directed $\to$ or $\uparrow$ in the plane. Such graphs can be recognized effectively (Felsner et al.). A 3DORG is the intersection graph of ...
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        252 views

        Why does Loday call the permutohedra “zylchgons”?

        Today I was reading Jean-Louis Loday's classic paper, "Realization of the Stasheff polytope", in which he produces a simple and very pretty realization of the associahedra as convex polytopes. He ...
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        3answers
        176 views

        Lattice points in a square pairwise-separated by integer distances

        Let $S_n$ be an $n \times n$ square of lattice points in $\mathbb{Z}^2$. Q1. What is the largest subset $A(n)$ of lattice points in $S_n$ that have the property that every pair of points in $A(n)$...
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        25 views

        Inefficient covering by translates

        While trying to answer this question, I arrived at another question: How many translates of $\{0,1\}^n$ does it take to cover $\mathbb F_3^n$? The broader context is: consider a set $S$ and a ...
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        22 views

        condition on rational polyhedral cone to guarantee dual cone is homogeneous

        Let $\sigma\subseteq \Bbb R^d$ be a full-dimensional rational polyhedral cone which is strongly convex (i.e. $\sigma\cap-\sigma=0$). Definition. The cone $\sigma$ is homogeneous if there are ...
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        70 views

        Random walks in arrangements of lines in the plane

        Let $\cal{A}_n$ be a simple arrangement of $n$ lines in $\mathbb{R}^2$. (Simple: each pair of lines meet in a distinct point, i.e., no three lines pass through the same point.) Start a random walk at ...
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        0answers
        139 views

        Can we represent partitions by mutually parallel lines in the plane?

        Lately I have become interested in the following idea: Suppose $n$ is a positive integer and $[n]=\{1,2,3,...,n\}$. Suppose we have 3 distinct partitions $b$, $g$, and $r$ of $[n]$. Assume that the ...
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        1answer
        267 views

        Can the graph of a symmetric polytope have more symmetries than the polytope itself?

        I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
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        2answers
        950 views

        Is it possible that both a graph and its complement have small connectivity?

        Let $G=(V,E)$ be a simple graph with $n$ vertices. The isoperimetric constant of $G$ is defined as $$ i(G) := \min_{A \subset V,|A| \leq \frac n2} \frac{|\partial A|}{|A|} $$ where $\partial A$ is ...
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        0answers
        42 views

        Upper bounds on $\epsilon$-covers of arbitrary compact manifolds

        Let $M \subset \mathbb{R}^d$ be a compact $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every ...
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        3answers
        210 views

        monochromatic subset

        Suppose we have $n^2$ red points and $n(n-1)$ blue points in the plane in general position. Is it possible to find a subset $S$ of red points such that the convex hull of $S$ does not contain any blue ...
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        1answer
        76 views

        Combinatorial problem about binary arrays with certain mutual distinctions

        If there are m binary arrays (with 0 and 1) of length n, and between any two of these m arrays, there are k and only k same numbers (with the same site index in two different arrays). For example, if ...

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