Triangles of Extremal Area or Perimeter in a Finite Planar Point Set

Braβ, P.; Rote, Günter; Swanepoel, Konrad J.

doi:10.1007/s00454-001-0010-6

Cited by 14 publications

(17 citation statements)

References 22 publications

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“…In particular, these rules imply that points selected during a single round of the dot puzzle must be non-decreasing, and thus at most 2n−3 points can be selected take part in Q i . Thus Similarly, we can almost recover the result ex(n, { , , , }) ≤ n of Braß, Rote and Swanepoel [3]. Here, the rules are:…”

Section: Some Warm-up Exercisessupporting

confidence: 55%

“…1 F:F n T2: [3] n T2:H n T2:H n T2:H n T2:H n T2:H n T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n T2: [3] n T2: [3] n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n : n 2 T2:H n T2: [3] n T2: [3] n T2:H n T2:H n T2:H n T2:H n T2:H n H:H n : n 2 T2: [2] n : n 2 T2: [2] n : n 2 T2: [2] n : n 2 T2: [2] n : n 2 T2: [2] n : n 2 T2: [2] n : n 2 T2: [2] n 2 [2]: [2] n T2: [3] n [3]: [3] n : n 2 T2: [2] n : n 2 [3]: [2] n : n 2 T2: [2] n : n 2 [3]: [2] n : n 2 T2: [2] n 2 [2]: [2] n : n 2 T2: [2] n : n 2 [3]: [2] n : n 2 T2: [2] n : n 2 [3]: [2] n : n 2 T2: [2] n : n 2 [3]: [2] n : n 2 T2: [2] [2] n : n 2 T2: [2] n : n 2 T2: [2] n : n 2 T2: [2] n : n 2 T2: [2] n : n 2 T2: [2] n : n 2 T2: [2] n 2 [...…”

Section: Previous Workmentioning

confidence: 99%

See 1 more Smart Citation

More Turán-Type Theorems for Triangles in Convex Point Sets

Aronov¹,

Dujmović

Morin

et al. 2019

Electron. J. Combin.

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We study the following family of problems: Given a set of n points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a log n factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.

show abstract

Section: Some Warm-up Exercisessupporting

confidence: 55%

Section: Previous Workmentioning

confidence: 99%

More Turán-Type Theorems for Triangles in Convex Point Sets

Aronov¹,

Dujmović

Morin

et al. 2019

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…The currently best upper bound, O(n 44/19 ), due to Dumitrescu, Sharir and C. Tóth [15], recently improved an older O(n 7/3 ) bound of Pach and Sharir [31]. Answering further questions of Erdős and Purdy [23], Braß, Rote and Swanepoel [9] showed the following two results.…”

Section: Introductionmentioning

confidence: 95%

On the Number of Tetrahedra with Minimum, Unit, and Distinct Volumes in Three-Space

Dumitrescu

Tóth

2008

Combinator. Probab. Comp.

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We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3-space, and in general in d dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by n points in R 3 is at most 2 3 n 3 − O(n 2 ), and there are point sets for which this number is 3 16 n 3 − O(n 2 ). We also present an O(n 3 ) time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for every k, d ∈ N, 1 ≤ k ≤ d, the maximum number of k-dimensional simplices of minimum (nonzero) volume spanned by n points in R d is Θ(n k ). (ii) The number of unit-volume tetrahedra determined by n points in R 3 is O(n 7/2 ), and there are point sets for which this number is Ω(n 3 log log n). (iii) For every d ∈ N, the minimum number of distinct volumes of all full-dimensional simplices determined by n points in R d , not all on a hyperplane, is Θ(n).

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“…The number of triangles with some extremal property might go up (significantly) when one moves up one dimension. For instance, Braß, Rote, and Swanepoel [9] have shown that the number of maximum area triangles in the plane is at most n (which is tight). In 3-space we show that this number is at least Ω(n 4/3 ) in the worst case.…”

mentioning

confidence: 97%

Extremal problems on triangle areas in two and three dimensions

Dumitrescu

Sharir

Tóth

2009

Journal of Combinatorial Theory, Series A

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The study of extremal problems on triangle areas was initiated in a series of papers by Erd\H{o}s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles. In the plane, our main result is an $O(n^{44/19}) =O(n^{2.3158})$ upper bound on the number of unit-area triangles spanned by $n$ points, which is the first breakthrough improving the classical bound of $O(n^{7/3})$ from 1992. We also make progress in a number of important special cases: We show that (i) For points in convex position, there exist $n$-element point sets that span $\Omega(n\log n)$ triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by $n$ points is at most ${2/3}(n^2-n)$; there exist $n$-element point sets (for arbitrarily large $n$) that span $(6/\pi^2-o(1))n^2$ minimum-area triangles. (iii) The number of acute triangles of minimum area determined by $n$ points is O(n); this is asymptotically tight. (iv) For $n$ points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are $n$-element point sets that span $\Omega(n\log n)$ minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds). In 3-space we prove an $O(n^{17/7}\beta(n))= O(n^{2.4286})$ upper bound on the number of unit-area triangles spanned by $n$ points, where $\beta(n)$ is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, $O(n^{8/3})$, is an old result from 1971.Comment: title page + 27 pages, 5 figure

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Triangles of Extremal Area or Perimeter in a Finite Planar Point Set

Cited by 14 publications

References 22 publications

More Turán-Type Theorems for Triangles in Convex Point Sets

More Turán-Type Theorems for Triangles in Convex Point Sets

On the Number of Tetrahedra with Minimum, Unit, and Distinct Volumes in Three-Space

Extremal problems on triangle areas in two and three dimensions

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