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.

KW - Antipodal pair

KW - Balancing

KW - Barycenter

KW - Borsuk-Ulam theorem

KW - Minkowski sum

KW - Skeleton of polyhedra

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 -