Mass Partitions via Equivariant Sections of Stiefel Bundles

We consider a geometric combinatorial problem naturally associated to the geometric topology of certain spherical space forms. Given a collection of $m$ mass distributions on $\mathbb{R}^n$, the existence of $k$ affinely independent regular $q$-fans, each of which equipartitions each of the measures, can in many cases be deduced from the existence of a $\mathbb{Z}_q$-equivariant section of the Stiefel bundle $V_k(\mathbb{F}^n)$ over $S(\mathbb{F}^n)$, where $V_k(\mathbb{F}^n)$ is the Stiefel manifold of all orthonormal $k$-frames in $\mathbb{F}^n,\, \mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, and $S(\mathbb{F}^n)$ is the corresponding unit sphere. For example, the parallelizability of $\mathbb{R}P^n$ when $n = 2,4$, or $8$ implies that any two masses on $\mathbb{R}^n$ can be simultaneously bisected by each of $(n-1)$ pairwise-orthogonal hyperplanes, while when $q=3$ or 4, the triviality of the circle bundle $V_2(\mathbb{C}^2)/\mathbb{Z}_q$ over the standard Lens Spaces $L^3(q)$ yields that for any mass on $\mathbb{R}^4$, there exist a pair of complex orthogonal regular $q$-fans, each of which equipartitions the mass.Comment: 11 pages, final versio

Equipartitioning triangles

An intriguing conjecture of Nandakumar and Ramana Rao is that for every convex body K ⊆ R2, and for any positive integer n, K can be expressed as the union of n convex sets with disjoint interiors and each having the same area and perimeter. The first difficult case- n = 3- was settled by Bárány, Blagojevi¢, and Szucs using powerful tools from algebra and equivariant topology. Here we give an elementary proof of this result in case K is a triangle, and show how to extend the approach to prove that the conjecture is true for triangles.Ministerio de Educación y CienciaEuropean Science FoundationNational Science Foundatio

Balanced Islands in Two Colored Point Sets in the Plane

Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $\alpha \in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil \alpha r\rceil$ red points and exactly $\lceil \alpha b \rceil$ blue points of $S$; a convex set containing exactly $\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an $O(n^4)$ time algorithm and an $O(n^2\log n)$ time algorithm in the second case. Finally, if $\lceil \alpha r\rceil+\lceil \alpha b\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$