Year: 2015
Type: article
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, that is, whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any compact planar set) 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 compact set containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) For any $d$-dimensional bounded convex polyhedron containing the origin, there exists a pair of antipodal points consisting of a point on a $\lfloor d/2 \rfloor$-face and a point on a $\lceil d/2\rceil$-face.
Institutions: ETH Zurich, Institute of Theoretical Computer Science, Korea Advanced Institute of Science and Technology, Binghamton University, Heinrich Heine University Düsseldorf, Kyoto University, Siemens (United States), University of Electro-Communications, HUN-REN Alfréd Rényi Institute of Mathematics, Fudan University, Kwansei Gakuin University, Moscow Institute of Physics and Technology, Carleton University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funders: National Science Foundation, Austrian Science Fund, Ministry of Education and Science of the Russian Federation, Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, Natural Sciences and Engineering Research Council of Canada, Shanghai Leading Academic Discipline Project, National Research, Development and Innovation Office
OpenAlex ID: W4298835865