Discrete & Computational Geometry
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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 $$S\subseteq
{\mathbb {R}}^d$$ and with $$s_0\in S$$ is almost-monochromatic if $$S\setminus
\{s_0\}$$ is monochromatic but S is not. We consider questions about finding
almost-monochromatic similar copies of pairs $$(S,s_0)$$ in colourings of
$${\mathbb {R}}^d$$ , $${\mathbb {Z}}^d$$ , and of $${\mathbb {Q}}$$ under some
restrictions ...
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 quadratic
functions is piecewise linear, and that all three functions have the same
critical points a...
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 the context of edge-isoperimetric inequalities on graphs, and it can
be derived from the known results that there is a linear order on the set of
n-tuples w...
Weakly Regular Subdivisions
Discrete & Computational Geometry - Tập 47 - Trang 106-116 - 2011
It is shown that 2-dimensional subdivisions can be made regular by moving their
vertices within parallel 1-dimensional spaces. As a consequence, any
2-dimensional subdivision is projected from the boundary complex of a
4-polytope.
Tổng số: 2,025
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