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

RESUMO

Makeev [2] stated that every finite Borel measure in $\mathbb R^d$ assigning zero mass to hyperplanes can be cut by $d$ mutually orthogonal hyperplanes so that every pair divides the measure into four equal parts, and outlined a proof strategy, but the key steps were left incomplete. We give a direct and elementary proof. The main ingredient is a Borsuk--Ulam theorem for $O(k)$: every $B_k$-equivariant map from $O(k)$ to a natural representation of the hyperoctahedral group $B_k$ has a zero. An explicit model map has one free orbit of zeros, consisting of the signed eigenbases of a diagonal operator with simple spectrum. A derivative computation and mod-$2$ equivariant degree complete the proof. This is a self-contained proof of a special case of the general Stiefel-manifold theorem in [5].