Weight balancing on boundaries and Skeletons

Luis Barba; Otfried Cheong; Jean Lou De Carufel; Michael Gene Dobbins; Rudolf Fleischer; Akitoshi Kawamura; Matias Korman; Yoshio Okamoto; János Pach; Yuan Tang; Takeshi Tokuyama; Sander Verdonschot; Tianhao Wang

doi:10.1145/2582112.2582142

Weight balancing on boundaries and Skeletons

Luis Barba, Otfried Cheong, Jean Lou De Carufel, Michael Gene Dobbins, Rudolf Fleischer, Akitoshi Kawamura, Matias Korman, Yoshio Okamoto, János Pach, Yuan Tang, Takeshi Tokuyama, Sander Verdonschot, Tianhao Wang

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

5 Citations (Scopus)

Abstract

Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.

Original language	English
Title of host publication	Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014
Publisher	Association for Computing Machinery
Pages	436-443
Number of pages	8
ISBN (Print)	9781450325943
DOIs	https://doi.org/10.1145/2582112.2582142
Publication status	Published - 2014
Externally published	Yes
Event	30th Annual Symposium on Computational Geometry, SoCG 2014 - Kyoto, Japan Duration: Jun 8 2014 → Jun 11 2014

Publication series

Name	Proceedings of the Annual Symposium on Computational Geometry

Other

Other	30th Annual Symposium on Computational Geometry, SoCG 2014
Country/Territory	Japan
City	Kyoto
Period	6/8/14 → 6/11/14

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Geometry and Topology
Computational Mathematics

Access to Document

10.1145/2582112.2582142

Cite this

Barba, L., Cheong, O., De Carufel, J. L., Dobbins, M. G., Fleischer, R., Kawamura, A., Korman, M., Okamoto, Y., Pach, J., Tang, Y., Tokuyama, T., Verdonschot, S., & Wang, T. (2014). Weight balancing on boundaries and Skeletons. In Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014 (pp. 436-443). (Proceedings of the Annual Symposium on Computational Geometry). Association for Computing Machinery. https://doi.org/10.1145/2582112.2582142

Weight balancing on boundaries and Skeletons. / Barba, Luis; Cheong, Otfried; De Carufel, Jean Lou et al.
Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014. Association for Computing Machinery, 2014. p. 436-443 (Proceedings of the Annual Symposium on Computational Geometry).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Barba, L, Cheong, O, De Carufel, JL, Dobbins, MG, Fleischer, R, Kawamura, A, Korman, M, Okamoto, Y, Pach, J, Tang, Y, Tokuyama, T, Verdonschot, S & Wang, T 2014, Weight balancing on boundaries and Skeletons. in Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014. Proceedings of the Annual Symposium on Computational Geometry, Association for Computing Machinery, pp. 436-443, 30th Annual Symposium on Computational Geometry, SoCG 2014, Kyoto, Japan, 6/8/14. https://doi.org/10.1145/2582112.2582142

@inproceedings{38a95a07abfb45368a5d48321aae36c0,

title = "Weight balancing on boundaries and Skeletons",

abstract = "Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.",

author = "Luis Barba and Otfried Cheong and {De Carufel}, {Jean Lou} and Dobbins, {Michael Gene} and Rudolf Fleischer and Akitoshi Kawamura and Matias Korman and Yoshio Okamoto and J{\'a}nos Pach and Yuan Tang and Takeshi Tokuyama and Sander Verdonschot and Tianhao Wang",

year = "2014",

doi = "10.1145/2582112.2582142",

language = "English",

isbn = "9781450325943",

series = "Proceedings of the Annual Symposium on Computational Geometry",

publisher = "Association for Computing Machinery",

pages = "436--443",

booktitle = "Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014",

note = "30th Annual Symposium on Computational Geometry, SoCG 2014 ; Conference date: 08-06-2014 Through 11-06-2014",

}

TY - GEN

T1 - Weight balancing on boundaries and Skeletons

AU - Barba, Luis

AU - Cheong, Otfried

AU - De Carufel, Jean Lou

AU - Dobbins, Michael Gene

AU - Fleischer, Rudolf

AU - Kawamura, Akitoshi

AU - Korman, Matias

AU - Okamoto, Yoshio

AU - Pach, János

AU - Tang, Yuan

AU - Tokuyama, Takeshi

AU - Verdonschot, Sander

AU - Wang, Tianhao

PY - 2014

Y1 - 2014

N2 - Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.

AB - Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.

UR - http://www.scopus.com/inward/record.url?scp=84904435691&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904435691&partnerID=8YFLogxK

U2 - 10.1145/2582112.2582142

DO - 10.1145/2582112.2582142

M3 - Conference contribution

AN - SCOPUS:84904435691

SN - 9781450325943

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 436

EP - 443

BT - Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014

PB - Association for Computing Machinery

T2 - 30th Annual Symposium on Computational Geometry, SoCG 2014

Y2 - 8 June 2014 through 11 June 2014

ER -

Weight balancing on boundaries and Skeletons

Abstract

Publication series

Other

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this