Combinatorial Distance Geometry in Normed Spaces

Swanepoel, Konrad J.

doi:10.1007/978-3-662-57413-3_17

Cited by 19 publications

(12 citation statements)

References 189 publications

(229 reference statements)

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“…Most of the results are inclusions between classes of intersection graphs of one-dimensional objects such as line segments and curves. A survey by Swanepoel [22] summarizes results on minimum distance graphs and unit distance graphs in normed spaces, including bounds on the minimum/maximum degree, maximum number of edges, chromatic number, and independence number.…”

Section: Other Related Workmentioning

confidence: 99%

Classifying Convex Bodies by their Contact and Intersection Graphs

Aamand¹,

Abrahamsen²,

Knudsen³

et al. 2019

Preprint

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Suppose that A is a convex body in the plane and that A1, . . . , An are translates of A. Such translates give rise to an intersection graph of A, G = (V, E), with vertices V = {1, . . . , n} and edges E = {uv | Au ∩ Av = ∅}. The subgraph G = (V, E ) satisfying that E ⊂ E is the set of edges uv for which the interiors of Au and Av are disjoint is a unit distance graph of A. If furthermore G = G, i.e., if the interiors of Au and Av are disjoint whenever u = v, then G is a contact graph of A.In this paper we study which pairs of convex bodies have the same contact, unit distance, or intersection graphs. We say that two convex bodies A and B are equivalent if there exists a linear transformation B of B such that for any slope, the longest line segments with that slope contained in A and B , respectively, are equally long. For a broad class of convex bodies, including all strictly convex bodies and linear transformations of regular polygons, we show that the contact graphs of A and B are the same if and only if A and B are equivalent. We prove the same statement for unit distance and intersection graphs.

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Section: Other Related Workmentioning

confidence: 99%

Classifying Convex Bodies by their Contact and Intersection Graphs

Aamand¹,

Abrahamsen²,

Knudsen³

et al. 2019

Preprint

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show abstract

Section: Lemma 31mentioning

confidence: 99%

Large Equilateral Sets in Subspaces of $$\ell _\infty ^n$$ of Small Codimension

Франкл

2021

Discrete Comput Geom

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For fixed k we prove exponential lower bounds on the equilateral number of subspaces of $$\ell _{\infty }^n$$ ℓ ∞ n of codimension k. In particular, we show that subspaces of codimension 2 of $$\ell _{\infty }^{n+2}$$ ℓ ∞ n + 2 and subspaces of codimension 3 of $$\ell _{\infty }^{n+3}$$ ℓ ∞ n + 3 have an equilateral set of cardinality $$n+1$$ n + 1 if $$n\ge 7$$ n ≥ 7 and $$n\ge 12$$ n ≥ 12 respectively. Moreover, the same is true for every normed space of dimension n, whose unit ball is a centrally symmetric polytope with at most $${4n}/{3}-o(n)$$ 4 n / 3 - o ( n ) pairs of facets.

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“…As far as we know, this problem has not been well studied in this space. Swanepoel [Swa18] showed that e(X ⊕ ∞ Y ) ≤ e(X)b f (Y ) for normed spaces X and Y , where b f is the finite Borsuk number. However, explicit bounds are still unknown when X and Y are Euclidean spaces and when we take an ℓ p sum instead of an ℓ ∞ sum.…”

Section: Previous Work and Our Resultsmentioning

confidence: 99%

Few distance sets in $\ell_p$ spaces and $\ell_p$ product spaces

Chen,

Gui,

Tang

et al. 2020

Preprint

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Kusner asked if n + 1 points is the maximum number of points in R n such that the ℓp distance between any two points is 1. We present an improvement to the best known upper bound when p is large in terms of n, as well as a generalization of the bound to s-distance sets. We also study equilateral sets in the ℓp sums of Euclidean spaces, deriving upper bounds on the size of an equilateral set for when p = ∞, p is even, and for any 1 ≤ p < ∞.

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Combinatorial Distance Geometry in Normed Spaces

Cited by 19 publications

References 189 publications

Classifying Convex Bodies by their Contact and Intersection Graphs

Classifying Convex Bodies by their Contact and Intersection Graphs

Large Equilateral Sets in Subspaces of $$\ell _\infty ^n$$ of Small Codimension

Few distance sets in $\ell_p$ spaces and $\ell_p$ product spaces

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