The number of double-normals in space

Kupavskii, Andrey

doi:10.1007/s00454-016-9804-4

Cited by 4 publications

(2 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that many different classes of dense geometric graphs were studied from a similar perspective. We mention diameter graphs [27,16,18] and double-normal graphs [22,23,17]. In some cases, the relationship between the largest clique and the maximum number of parts in an arbitrarily large complete multipartite graph is quite intricate, as it is the case for double-normal graphs, see [17].…”

Section: Resultsmentioning

confidence: 99%

Nearly $k$-distance sets

Франкл¹,

Kupavskii²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We say that a set of points 𝑆 ⊂ R 𝑑 is an 𝜀-nearly 𝑘-distance set if there exist 1 ≤ 𝑡 1 ≤ . . . ≤ 𝑡 𝑘 , such that the distance between any two distinct points in 𝑆 falls intoIn this paper, we study the quantityand its relation to the classical quantity 𝑚 𝑘 (𝑑): the size of the largest 𝑘-distance set in R 𝑑 . We obtain that 𝑀 𝑘 (𝑑) = 𝑚 𝑘 (𝑑) for 𝑘 = 2, 3, as well as for any fixed 𝑘, provided that 𝑑 is sufficiently large. The last result answers a question, proposed by Erdős, Makai and Pach.We also address a closely related Turán-type problem, studied by Erdős, Makai, Pach, and Spencer in the 80's: given 𝑛 points in R 𝑑 , how many pairs of them form a distance that belongs towhere 𝑡 1 , . . . , 𝑡 𝑘 are fixed and any two points in the set are at distance at least 1 apart? We establish the connection between this quantity and a quantity closely related to 𝑀 𝑘 (𝑑 − 1), as well as obtain an exact answer for the same ranges 𝑘, 𝑑 as above.

show abstract

Section: Resultsmentioning

confidence: 99%

Nearly $k$-distance sets

Франкл¹,

Kupavskii²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We note that many different classes of dense geometric graphs were studied from a similar perspective. We mention diameter graphs [ 19 , 21 , 28 ] and double-normal graphs [ 20 , 24 , 25 ]. In some cases, the relationship between the largest clique and the maximum number of parts in an arbitrarily large complete multipartite graph is quite intricate, as it is the case for double-normal graphs, see [ 20 ].…”

Section: Introductionmentioning

confidence: 99%

Nearly k-Distance Sets

Франкл

Kupavskii

2023

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

We say that a set of points $$S\subset {{\mathbb {R}}}^d$$ S ⊂ R d is an $$\varepsilon $$ ε -nearly k-distance set if there exist $$1\le t_1\le \ldots \le t_k$$ 1 ≤ t 1 ≤ … ≤ t k , such that the distance between any two distinct points in S falls into $$[t_1,t_1+\varepsilon ]\cup \cdots \cup [t_k,t_k+\varepsilon ]$$ [ t 1 , t 1 + ε ] ∪ ⋯ ∪ [ t k , t k + ε ] . In this paper, we study the quantity $$\begin{aligned} M_k(d) = \lim _{\varepsilon \rightarrow 0}\max {\{|S|:S\,\text { is an}\, \varepsilon \text {-nearly}\, k\text {-distance set in}\,{{\mathbb {R}}}^d\}} \end{aligned}$$ M k ( d ) = lim ε → 0 max { | S | : S is an ε -nearly k -distance set in R d } and its relation to the classical quantity $$m_k(d)$$ m k ( d ) : the size of the largest k-distance set in $${{\mathbb {R}}}^d$$ R d . We obtain that $$M_k(d)=m_k(d)$$ M k ( d ) = m k ( d ) for $$k=2,3$$ k = 2 , 3 , as well as for any fixed k, provided that d is sufficiently large. The last result answers a question, proposed by Erdős, Makai, and Pach. We also address a closely related Turán-type problem, studied by Erdős, Makai, Pach, and Spencer in the 90s: given n points in $${{\mathbb {R}}}^d$$ R d , how many pairs of them form a distance that belongs to $$[t_1,t_1+1]\cup \cdots \cup [t_k,t_k+1]$$ [ t 1 , t 1 + 1 ] ∪ ⋯ ∪ [ t k , t k + 1 ] , where $$t_1,\dots ,t_k$$ t 1 , ⋯ , t k are fixed and any two points in the set are at distance at least 1 apart? We establish the connection between this quantity and a quantity closely related to $$M_k(d-1)$$ M k ( d - 1 ) , as well as obtain an exact answer for the same ranges k, d as above.

show abstract

Acute Sets of Exponentially Optimal Size

Gerencsér

Harangi

2018

Discrete Comput Geom

View full text Add to dashboard Cite

We present a simple construction of an acute set of size 2 d−1 + 1 in R d for any dimension d. That is, we explicitly give 2 d−1 + 1 points in the d-dimensional Euclidean space with the property that any three points form an acute triangle. It is known that the maximal number of such points is less than 2 d . Our result significantly improves upon a recent construction, due to Dmitriy Zakharov, with size of order ϕ d where ϕ = (1 + √ 5)/2 ≈ 1.618 is the golden ratio.2010 Mathematics Subject Classification. 51M04, 51M15.

show abstract

The number of double-normals in space

Cited by 4 publications

References 9 publications

Nearly $k$-distance sets

Nearly $k$-distance sets

Nearly k-Distance Sets

Acute Sets of Exponentially Optimal Size

Contact Info

Product

Resources

About