Classification of Closed Minimal Networks on Flat Two-Dimensional Tori

Ivanov, A O; Ptitsyna, I V; Tuzhilin, A. A.

doi:10.1070/sm1994v077n02abeh003448

Cited by 11 publications

(2 citation statements)

References 9 publications

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“…A triangular pyramid is called an isosceles tetrahedron or a disphenoid, if all its faces are pairwise congruent. It is well-known that the surface of an isosceles tetrahedron can be locally isometric branched covered by the Euclidean plane, see [18]. Applying Corollary 3.5 we obtain the following result.…”

Section: Examples: Polyhedra and Conesmentioning

confidence: 72%

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Branched coverings and Steiner ratio

Ivanov

Tuzhilin

2015

Int Trans Operational Res

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For a branched locally isometric covering of metric spaces with intrinsic metrics, it is proved that the Steiner ratio of the base is not less than the Steiner ratio of the total space of the covering. As applications, it is shown that the Steiner ratio of the surface of an isosceles tetrahedron is equal to the Steiner ratio of the Euclidean plane, and that the Steiner ratio of a flat cone with angle of 2π/k at its vertex is also equal to the Steiner ratio of the Euclidean plane.

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Section: Examples: Polyhedra and Conesmentioning

confidence: 72%

“…It is well‐known that the surface of an isosceles tetrahedron can be locally isometric branched covered by the Euclidean plane, see Ivanov et al. (). Applying Corollary we obtain the following result.…”

Section: Examples: Polyhedra and Conesmentioning

confidence: 99%

Branched coverings and Steiner ratio

Ivanov

Tuzhilin

2015

Int Trans Operational Res

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Minimal Networks on Balls and Spheres for Almost Standard Metrics

Sciaraffia

2024

J Geom Anal

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We study the existence of minimal networks in the unit sphere $${\textbf{S}}^d$$ S d and the unit ball $${\textbf{B}}^d$$ B d of $${\textbf{R}}^d$$ R d endowed with Riemannian metrics close to the standard ones. We employ a finite-dimensional reduction method, modelled on the configuration of $$\theta $$ θ -networks in $${\textbf{S}}^d$$ S d and triods in $${\textbf{B}}^d$$ B d , jointly with the Lusternik–Schnirelmann category.

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