Convex Equipartitions via Equivariant Obstruction Theory (1202.5504v3)

Abstract: We describe a regular cell complex model for the configuration space F(\R^d,n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.

Convex Equipartitions via Equivariant Obstruction Theory

The paper presents a significant advancement in the domain of geometric measure theory, focusing on the conjecture proposed by Nandakumar and Ramana Rao. The authors apply Equivariant Obstruction Theory to validate the conjecture's prime power case. The conjecture posits that any convex polygon can be partitioned into $n$ convex segments, each with an identical area and perimeter.

The authors leverage a cell complex model for the configuration space $F(\mathbb{R}^2;n)$. Through exploration of the space, in conjunction with Equivariant Obstruction Theory, they arrive at a proof for cases where $n$ is a power of a prime number. This is accomplished by analyzing the associated cell complexes and proving the non-existence of $S_n$-equivariant maps from these configuration spaces to specific spheres.

Their approach addresses the inability to use Euler class arguments on non-compact manifolds directly. Instead, by constructing a sophisticated $S_n$-equivariant cell complex model for $F(\mathbb{R}^2;n)$, the paper establishes the correspondence between equipartition problems and configuration spaces.

Significantly, the work extends to $d$-dimensional generalizations, where it considers partitions of $d$-dimensional polytopes or measures on $\mathbb{R}^d$ into $n$ convex pieces with equalized functions. They confirm the existence of such partitions when $n$ is a prime power and report that the opposite holds true when $n$ is not a prime power.

A vital outcome from this research is Theorem 1.2, which asserts that an $S_n$-equivariant map from $F(\mathbb{R}^2;n)$ to $S_n \backslash S(W^{\otimes(d-1)})$ exists only if $n$ is not a prime power. The authors construct a beautiful $S_n$-equivariant (d-1)(n-1)-dimensional cell complex as part of their methodological contributions.

Critical numerical results involve the demonstrations of primary obstructions for mapping from these cell complexes to equivariant spheres. Notably, explicit calculation shows that the equivariant map exists if and only if $$gcd\left(\binom{n}{1}, \binom{n}{2}, \ldots, \binom{n}{n-1}\right) = 1$$, a condition derived from the classical results of Ram, and addressing the binomial coefficients' gcd.

The theoretical implications are profound, offering a broader perspective on analyzing symmetric configuration spaces through an algebraic topological lens, potentially influencing computational geometry and partitioning algorithms. Practically, while applications are more indirect, the mathematical structures and techniques developed could inform algorithms in computational geometry and related measures, providing rigorous underpinnings for exploratory and optimization problems.

Future research directions could involve extending these proofs and methodologies to broader classes of figures and higher-dimensional spaces, as well as exploring other algebraic invariants within topological and geometrical structures, further paving links between pure and applied mathematics.