Borsuk-Ulam type theorem for the orthogonal group and orthogonal four-partitions

Source abstract

Given a finite Borel measure $μ$ in $\mathbb{R}^d$, when can one find $d$ mutually orthogonal hyperplanes such that \emph{every pair} of them cuts $μ$ into four equal parts? Makeev [2] stated this result and outlined a proof strategy, but the key steps were left incomplete. We give the first complete proof. The key step is a Borsuk--Ulam-type theorem for the orthogonal group~$O(k)$: every continuous equivariant map from~$O(k)$ to a certain representation of the hyperoctahedral group~$B_k$ must vanish somewhere. We construct an explicit model map whose zero set consists of exactly one free $B_k$-orbit --- the set of all signed eigenbases of a fixed generic self-adjoint operator~$A$ --- verify nondegeneracy by an explicit derivative calculation, and conclude by the equivariant degree principle. The proof requires only linear algebra and elementary topology. The four-partition theorem follows immediately: the orthogonal hyperplanes are encoded as a frame in $O(d)$, and the equivariant map records the imbalance of $μ$ across each pair of hyperplanes. A zero of this map is the desired configuration. The result is a special case of a general zero theorem for Stiefel manifolds proved in [5] by different methods.