Discrete & Computational Geometry
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On the coordinatization of oriented matroids
Discrete & Computational Geometry - Tập 1 - Trang 293-306 - 1986
Several important and hard realizability problems of combinatorial geometry can be reduced to the realizability problem of oriented matroids. In this paper we describe a method to find a coordinatization for a large class of realizable cases. This algorithm has been used successfully to decide several geometric realizability problems. It is shown that all realizations found by our algorithm fulfil......
Reconstructing Plane Quartics from Their Invariants
Discrete & Computational Geometry - Tập 63 Số 1 - Trang 73-113 - 2020
k-Sets in Four Dimensions
Discrete & Computational Geometry - Tập 35 - Trang 177-191 - 2005
We show, with an elementary proof, that the number of halving simplices
in a set of n points in ℝ4 in general position is
O(n4-2/45). This improves the previous bound of
O(n4-1/13^{4}). Our main new ingredient is a bound on the maximum number of halving simplices intersecting a fixed 2-plane.
Almost-Monochromatic Sets and the Chromatic Number of the Plane
Discrete & Computational Geometry - Tập 70 - Trang 753-772 - 2023
In a colouring of
$${\mathbb {R}}^d$$
a pair
$$(S,s_0)$$
with
$$S\subseteq {\mathbb {R}}^d$$
a......
Densest packings of typical convex sets are not lattice-like
Discrete & Computational Geometry - Tập 14 - Trang 1-8 - 1995
We show that ifP is a convex polygon which has no parallel sides, then the densest packing of the plane with congruent copies ofP is not lattice-like. As a corollary we obtain that, in the sense of Baire categories, for most convex disks densest packing is not lattice-like.
Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics
Discrete & Computational Geometry - Tập 67 - Trang 811-842 - 2022
The Voronoi tessellation in
$${{{\mathbb {R}}}}^d$$
is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise......
Asymptotics for Some Combinatorial Characteristics of the Convex Hull of a Poisson Point Process in the Clifford Torus
Discrete & Computational Geometry - Tập 49 Số 2 - Trang 200-220 - 2013
Let $$\mathcal P _\lambda $$ be a homogeneous Poisson point process of rate $$\lambda $$ in the Clifford torus $$T^2\subset \mathbb E ^4$$ . Let $$(f_0, f_1, f_2, f_3)$$ be the $$f$$ -vector of conv $$\,\mathcal P _\lambda $$ and let $$\bar{v}$$ be the mean valence of a vertex of the convex hull. Asymptotic expressions for $$\mathsf E \, f_1$$ , $$\mathsf E \, f_2$$ , $$\mathsf E \, f_3$$ and $$\m......
A Partitioned Version of the Erdös–Szekeres Theorem for Quadrilaterals
Discrete & Computational Geometry - Tập 30 - Trang 321-336 - 2003
We prove a partitioned version of the Erdös–Szekeres theorem for the case
$k = 4$: any finite set $X \subset \bbbr^2$ of points in general position
can be
partitioned into sets $X_0, X_{ij}$ where $i=1,2,3,4$ and $j=1,\ldots,26$,
so that $|X_{1j}|=|X_{2j}|=|X_{3j}|=|X_{4j}|$, $|X_0|\leq 4$ and for all $j$
every transversal $\{x_1,x_2,x_3,x_4\}$,
$x_1 \in X_{1j}, x_2 \in X_{2j},x_3 \in X_{3j}, x_4......
Minimising the Sum of Projections of a Finite Set
Discrete & Computational Geometry - Tập 60 - Trang 493-511 - 2018
Consider the projections of a finite set
$$A\subset {\mathbb R}^n$$
onto the coordinate hyperplanes; how small can the sum of the sizes of these projections be, given the size of A? In a different form, this problem has been studied earlier in th......
Tổng số: 2,025
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