A lower bound on opaque sets

Akitoshi Kawamura; Sonoko Moriyama; Yota Otachi; János Pach

doi:10.1016/j.comgeo.2019.01.002

A lower bound on opaque sets

Akitoshi Kawamura, Sonoko Moriyama, Yota Otachi, János Pach

Mathematical Informatics

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Abstract

It is proved that the total length of any set of countably many rectifiable curves whose union meets all straight lines that intersect the unit square U is at least 2.00002. This is the first improvement on the lower bound of 2 known since 1964. A similar bound is proved for all convex sets U other than a triangle.

Original language	English
Pages (from-to)	13-22
Number of pages	10
Journal	Computational Geometry: Theory and Applications
Volume	80
DOIs	https://doi.org/10.1016/j.comgeo.2019.01.002
Publication status	Published - Jul 2019

All Science Journal Classification (ASJC) codes

Computer Science Applications
Geometry and Topology
Control and Optimization
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.comgeo.2019.01.002

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title = "A lower bound on opaque sets",

abstract = "It is proved that the total length of any set of countably many rectifiable curves whose union meets all straight lines that intersect the unit square U is at least 2.00002. This is the first improvement on the lower bound of 2 known since 1964. A similar bound is proved for all convex sets U other than a triangle.",

author = "Akitoshi Kawamura and Sonoko Moriyama and Yota Otachi and J{\'a}nos Pach",

note = "Funding Information: The work presented here was supported in part by the JSPS KAKENHI (Grant-in-Aid for Challenging Research (Exploratory)) JP17K19960; by the MEXT KAKENHI (Grants-in-Aid for Scientific Research on Innovative Areas) JP24106002, JP24106004, JP25106505 under the ELC project; by OTKA under EUROGIGA projects GraDR and ComPoSe 10-EuroGIGA-OP-003; and by Swiss National Science Foundation Grants 200020-162884 and 200021-165977. A preliminary version was presented at the 32nd Annual Symposium on Computational Geometry (SoCG 2016) [10]. We are grateful to G?bor Tardos for many interesting discussions on the subject. In particular, the present proof of Lemma 4 is based on his idea.",

year = "2019",

month = jul,

doi = "10.1016/j.comgeo.2019.01.002",

language = "English",

volume = "80",

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journal = "Computational Geometry: Theory and Applications",

issn = "0925-7721",

publisher = "Elsevier",

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AU - Kawamura, Akitoshi

AU - Moriyama, Sonoko

AU - Otachi, Yota

AU - Pach, János

N1 - Funding Information: The work presented here was supported in part by the JSPS KAKENHI (Grant-in-Aid for Challenging Research (Exploratory)) JP17K19960; by the MEXT KAKENHI (Grants-in-Aid for Scientific Research on Innovative Areas) JP24106002, JP24106004, JP25106505 under the ELC project; by OTKA under EUROGIGA projects GraDR and ComPoSe 10-EuroGIGA-OP-003; and by Swiss National Science Foundation Grants 200020-162884 and 200021-165977. A preliminary version was presented at the 32nd Annual Symposium on Computational Geometry (SoCG 2016) [10]. We are grateful to G?bor Tardos for many interesting discussions on the subject. In particular, the present proof of Lemma 4 is based on his idea.

PY - 2019/7

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