Bisection of measures on spheres and a fixed point theorem

Abstract

We establish a variant for spheres of results obtained in [7], [3] for affine space. The principal result, that, if $m$ is a power of $2$ and $k\geq 1$, then $km$ continuous densities on the unit sphere in $\mathbb R^{m+1}$ may be simultaneously bisected by a set of at most $k$ hyperplanes through the origin, is essentially equivalent to the main theorem of Hubard and Karasev in [7]. But the methods used, involving Euler classes of vector bundles over symmetric powers of real projective spaces and an 'orbifold' fixed point theorem for involutions, are substantially different from those in [7], [3].