BISECTION OF MEASURES ON SPHERES AND A FIXED POINT THEOREM.

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 ≥ 1, then km continuous densities on the unit sphere in 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]. [ABSTRACT FROM AUTHOR]