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      Questions tagged [discrete-geometry]

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

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      0answers
      42 views

      The scutoid as a noninscribable polyhedron

      I have no a good backround in combinatorial geometry but I would like to ask next question, because I think that it is interesting. I know from the book [1], section B18 and the post Combinatorially ...
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      0answers
      56 views

      Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?

      I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope. I am looking ...
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      1answer
      86 views

      Intersection of a lower dimensional space and a discrete set

      Let $H\subset \mathbb{R}^n$ with dimension ${\rm dim}(H)=\ell<n$; let $S$ be a finite subset of reals. My question is the following. Is it correct to say, $$ {\rm card}(H \cap V)\leqslant |S|^\...
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      1answer
      145 views

      On covering convex 2D regions with rectangles

      Given a convex 2D region $C$ and a positive integer $N$. We need to cover $C$ with $N$ rectangles such that the sum of the areas of the $N$ rectangles is the least – no further constraints on the ...
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      1answer
      58 views

      Large subsets of the Hamming cube with small intersections with all spheres of given radius

      What is the maximal cardinality of a subset $A$ of $\{-1,1\}^n$ such that any Hamming sphere with radius $r$ contains at most $k$ elements of $A$? Are explicit constructions with large cardinality ...
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      0answers
      68 views

      Maximum number of ways of splitting a set of points with an hyperplane

      Given a set $S$ of $n$ points in $\mathbb{R}^d$, let $D_S$ be the set $\{\mathbf{v}=|\mathbf{u}-\mathbf{u'}|: \mathbf{u},\mathbf{u'}\in S\}$ (where $\forall i=1,2,\ldots, d$, $\mathbf{v}_i=|\mathbf{u}...
      2
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      1answer
      35 views

      Polytopes and polyhedral cones in complex Euclidean space

      Given $A \in \mathsf{M}_{m \times n}(\mathbb{R})$ and $b \in \mathbb{R}^m$, the polyhedron with respect to $A$ and $b$, denoted by $P(A,b)$, is defined by $$ \{ x \in \mathbb{R}^n \mid Ax \le b \}.$$ ...
      3
      votes
      1answer
      60 views

      Regular triangulations of star-convex polyhedra with given boundary

      Given an $n$-dimensional star-convex polyhedron $P\subset \mathbb{R}^n$ with simplicial facets, is it always possible to construct a regular triangulation $K$ of $P$ which does not subdivide the ...
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      0answers
      45 views

      On non-convex polygons that tile convex polygons

      Polyominoes are rectilinear polygons - all angles are 90 or 270 degrees. It is well-known that among non-convex polyominoes are several whose copies can neatly tile rectangle. And for several such '...
      4
      votes
      1answer
      299 views

      Why does $\sqrt 5$ occur in manageable situations of these scenarios?

      Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7968198&tag=...
      6
      votes
      1answer
      129 views

      Point distributions in unit square which minimize E[1 / distance]

      Choose $n$ points $p_1,\ldots,p_n$ in the unit square $[0,1]^2\subset\mathbb{R}^2$ such that $D:=\mathop{\sum}\limits_{1\le i<j\le n}\frac{1}{dist(p_i,p_j)}$ is minimized, where $dist(p_i,p_j)$ is ...
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      0answers
      32 views

      Lattices with no roots and spread out shells

      I am looking for lattices with the following properties: The lattice has no roots. The norm (squared length) of the second shortest vectors should be at least twice as large as the norm of the ...
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      0answers
      120 views

      The drawn diagonals divide the $N\times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$

      Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N\times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N\times N$ board into $K$ regions. For each ...
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      0answers
      95 views

      Aperiodic tile with rational area

      Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having ...
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      3answers
      448 views

      Two queries on triangles, the sides of which have rational lengths

      Let us define a "rational triangle" as one in the Euclidean plane, with lengths of all sides rational. We are aware that a positive integer is called "congruent" only if it is the area of a right ...

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