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        Questions tagged [discrete-geometry]

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

        2
        votes
        1answer
        39 views

        Probabilities of Four Points Being in Convex/Deltoid Configurations

        Question: what is the probability that four distinct points in general position in the Euclidean plane are in convex configuration, depending on the number of leaf nodes in their Minimum ...
        2
        votes
        1answer
        60 views

        Diagonal shortcuts to minimize all-pairs shortest-paths in grid graph

        Augment the grid graph $G$ on lattice points $[1,n]^2$, which connects each point to its four distance-$1$ vertical and horizontal neighbors. Augment $G$ to $G'$ by adding in one of the two $\sqrt{2}$ ...
        4
        votes
        0answers
        69 views

        Discrepancy of the finite approximation of the Lebesgue measure

        Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...
        6
        votes
        2answers
        105 views

        Geometric dissection theory

        A few days ago, i realized that one way to prove the Pythagorean Theorem is to dissect the given right-angled triangle into 2 triangles similar to it, and apply well-known properties of ratios of ...
        3
        votes
        1answer
        136 views

        Sphere packing and kissing numbers in 3D

        When one looks at the way cannon balls and oranges are normally packed by the military and by groceries, it seems intuively clear that there is no way anybody can pack these any tighter. However, it ...
        2
        votes
        0answers
        43 views

        Optimal $f$-vector properties of translationally invariant 3-honeycombs for error correction of a photonic quantum computer

        In terms of the $f$-vector for a translationally invariant (in $\Bbb R^3$) honeycomb define $$ \begin{split} v &= \max\left( \frac{f_1}{f_0}, \frac{f_2}{f_3}\right), \\ f &= \max\left( \frac{...
        2
        votes
        1answer
        99 views

        Number of bitangents to convex polytopes

        Let me state my question prior to defining terms: Q. Let $P_1$ and $P_2$ be disjoint convex polytopes in $\mathbb{R}^d$ of $n$ vertices each. What is the maximum number of distinct bitangent ...
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        votes
        0answers
        58 views

        Minimize number of lattice paths below a given path

        Every north-east lattice path (NE-path) $v$ from $(0,0)$ to $(k, a)$ can be identified with a sequence $0 \le \lambda_1 \le \lambda_2 \le . . . \le \lambda_k\le a$, that represent the hight of each ...
        3
        votes
        0answers
        94 views

        Are triangulations with common refinements PL-homeomorphic?

        Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...
        3
        votes
        1answer
        57 views

        Tiling the surface of a hypersphere with regular simplices

        Let $S^{n-1} = \{x \in \mathbb{R}^n : x_1^2 + \cdots + x_n^2 = 1\}$. Consider a regular spherical simplex, obtained e.g. by taking a hyperspherical cap, picking $n$ equally-spaced points $P = \{p_1, \...
        3
        votes
        1answer
        248 views

        How to find the vertices of the set $\{v_i\in \mathbb{R}:a_1\ge v_1\ge v_2\ge \cdots\ge v_n\ge 0,\ q_2\le \sum_{i=1}^n p_iv_i\le q_1\}$

        I am given a set of inequalities $v_1\ge v_2\ge \cdots\ge v_n\ge 0$, $q_2\le \sum_{i=1}^n p_iv_i\le q_1$, with $\{p_i\}_{i=1}^n,\ q_1,q_2$ positive reals, and only one bound for the coordinates: $v_1\...
        4
        votes
        1answer
        223 views

        Graphs with adjacency matrix depending on associated-vector distances

        Let $G$ be a graph of order $n$ such that for each vertex $v$ there are two associated vectors, $f_v, g_v\in R^n$, where $uv\in E(G)$ if and only if $\|f_u - f_v\|^2 \ge \|g_u-g_v\|^2$. ISGCI didn't ...
        1
        vote
        0answers
        113 views

        Is there a method to cut a hypercube into disjoint cubes [closed]

        Since Borsuk conjecture hold for centrally symmetric convex sets in $\mathbb{R}^n$ so we can cut a hypercube into at least $n+1$ disjoint parts. Is there a method how can one do that?
        2
        votes
        0answers
        79 views

        A theory of (or reference for) symmetric point arrangements

        I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some ...
        0
        votes
        1answer
        116 views

        Reference request on Borsuk conjecture [closed]

        I just heard of Borsuk conjecture. I want to ask if there are any references preferably looking at the problem from the point of view of Mathematical analysis I can study it from? Thanks

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