Bisection of measures on spheres and a fixed point theorem

We establish a variant for spheres of results obtained in \cite{HK}, \cite{BBK} 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 \cite{HK}. 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 \cite{HK}, \cite{BBK}.