Регулярность минимайзеров функционала максимального расстояния

Я. И. Теплицкая

doi:10.1007/s10958-018-3866-1

Publications

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Регулярность минимайзеров функционала максимального расстояния

Записки научных семинаров ПОМИ РАН. 2017. Т. 462. С. 103–111.

Teplitskaya Y.

In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.

Language: Russian

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Keywords: regularity maximal distance minimizer Steiner tree locally minimal network дерево Штейнера локально минимальная сеть минимайзер максимального расстояния регулярность

PAGR: Accelerating Global Routing for VLSI Design Flow

Roman A. Solovyev, Mkrtchan I., Telpukhov D. et al., IEEE Access 2025 Vol. 13 P. 6440–6450

In this paper, we present PAGR (Python Alpha Global Routing) – a solution to the global routing problem in physical synthesis based on data from the ISPD 2024 contest. Our solution constructs a weighted graph and builds a Steiner tree. To accelerate the Steiner tree search, we propose a technique for the graph size minimization by reducing the ...

Added: April 9, 2025

Regular behaviour of the maximal hypergraph chromatic number

Cherkashin D., Petrov F., / Series arXiv "math". 2019.

Let m(n, r) denote the minimal number of edges in an n-uniform hypergraph which is not r-colorable. It is known that for a fixed n one has c_n r^n < m(n, r) < C_n r^n . We prove that for any fixed n the sequence a_r := m(n, r)/r^n has a limit, which was conjectured ...

Added: October 22, 2019

On regularity of maximal distance minimizers

Teplitskaya Y., / Cornell University. Серия arXiv "math". 2019.

We study the properties of sets Σ which are the solutions of the maximal distance minimizer problem, id est of sets having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ R 2 satisfying the inequality maxy∈M dist (y, Σ) ≤ r for a given compact set M ⊂ ...

Added: October 21, 2019

On the horseshoe conjecture for maximal distance minimizers

Danila Cherkashin, Yana Teplitskaya, ESAIM - Control, Optimisation and Calculus of Variations 2018 Vol. 24 No. 3 P. 1015–1041

We study the properties of sets Σ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ⊂R2satisfying the inequality maxy∈Mdist(y,Σ)≤r for a given compact set M⊂R2 and some given r>0. Such sets can be considered shortest possible pipelines arriving at a distance at most r to every point of M which in this case is considered as the set of customers of ...

Added: September 10, 2018

Regularity of the Monge-Ampère equation in Besov's space

Kolesnikov A., Tikhonov S. Y., / Series math "arxiv.org". 2012. No. 1203.3457.

Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi$ be the optimal transportation mapping pushing forward $\mu$ onto a log-concave compactly supported measure $\nu = e^{-W} \ dx$. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{\`e}re equation $e^{-V} = \det D^2 ...

Added: March 28, 2013