Smallest universal covers for families of triangles

Park, Ji Won; Cheong, Otfried

doi:10.1016/j.comgeo.2020.101686

Cited by 2 publications

(6 citation statements)

References 7 publications

(9 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Füredi and Wetzel showed that the same holds for the family of all triangles of diameter one [4], and Park and Cheong showed the same for the family of triangles of circumradius one, as well as for any family of two triangles [8]. These known results led Park and Cheong to make the following conjecture: Conjecture 1 ( [8]). For any bounded family T of triangles there is a triangle Z that is a smallest convex cover for T .…”

Section: Introductionmentioning

confidence: 88%

“…It is easy to see that this is equivalent to the following conjecture: Conjecture 2 ( [8]). Let X be a convex set.…”

Section: Introductionmentioning

confidence: 92%

“…Kovalev showed that the smallest convex cover for the family of all triangles of perimeter one is a uniquely determined triangle [7,4]. Füredi and Wetzel showed that the same holds for the family of all triangles of diameter one [4], and Park and Cheong showed the same for the family of triangles of circumradius one, as well as for any family of two triangles [8]. These known results led Park and Cheong to make the following conjecture: Conjecture 1 ( [8]).…”

Section: Introductionmentioning

confidence: 94%

“…It follows from the complete characterization of the smallest convex cover for two given triangles by Park and Cheong [8] that C is a smallest convex cover for T 0 and T θ0,θ0 . This makes C a smallest convex cover for the family of all triangles contained in the unit square.…”

Section: The Isosceles Case: Construction Of the Covermentioning

confidence: 99%

“…In our second main result, we consider Conjecture 2. It is known to hold when X is a disk [8], a half-disk (Theorem 3), or a square (Theorem 4). In Section 3, we prove the following theorem, which extends this to a much larger family of shapes X : Fig.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Covering families of triangles

et al. 2023

Self Cite

View full text Add to dashboard Cite

A cover for a family $${{\mathcal {F}}}$$ F of sets in the plane is a set into which every set in $${{\mathcal {F}}}$$ F can be isometrically moved. We are interested in the convex cover of smallest area for a given family of triangles. Park and Cheong conjectured that any family of triangles of bounded diameter has a smallest convex cover that is itself a triangle. The conjecture is equivalent to the claim that for every convex set $${{\mathcal {X}}}$$ X there is a triangle Z whose area is not larger than the area of $${{\mathcal {X}}}$$ X , such that Z covers the family of triangles contained in $${{\mathcal {X}}}$$ X . We prove this claim for the case where a diameter of $${{\mathcal {X}}}$$ X lies on its boundary. We also give a complete characterization of the smallest convex cover for the family of triangles contained in a half-disk, and for the family of triangles contained in a square. In both cases, this cover is a triangle.

show abstract

Section: Introductionmentioning

confidence: 88%

“…It is easy to see that this is equivalent to the following conjecture: Conjecture 2 ( [8]). Let X be a convex set.…”

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 94%

Section: The Isosceles Case: Construction Of the Covermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Covering families of triangles

et al. 2023

Self Cite

View full text Add to dashboard Cite

show abstract

Universal convex covering problems under translations and discrete rotations

Jung,

Yoon,

Ahn

et al. 2023

Advances in Geometry

View full text Add to dashboard Cite

We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently, closed curves of length 2) allowing translations and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translations and discrete rotations of π are allowed. We also give convex coverings of closed curves of length 2 under translations and discrete rotations of multiples of π/2 and of 2π/3. We show that no proper closed subset of that covering is a covering for discrete rotations of multiples of π/2, which is an equilateral triangle of height smaller than 1, and conjecture that such a covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translations and discrete rotations of 2π/k for all integers k=3.

show abstract

Smallest universal covers for families of triangles

Cited by 2 publications

References 7 publications

Covering families of triangles

Covering families of triangles

Universal convex covering problems under translations and discrete rotations

Contact Info

Product

Resources

About