Weight Balancing on Boundaries

TL;DR

This paper generalizes classical boundary point properties of polygons and polyhedra, proving new results about weight placement, equilateral triangles, and antipodal pairs in higher dimensions.

Contribution

It introduces three novel theorems extending boundary point properties to weighted points, 3D sets, and higher-dimensional convex polyhedra.

Findings

Weighted barycenter placement on polygon boundaries

Existence of equilateral triangles centered at the origin in 3D sets

Antipodal point pairs on convex polyhedra in higher dimensions

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.