Before revision:  This question was already studied in the plane, our contribution is to extend the planar results to higher dimensions . All of our results are proved using topological methods. We use some well-established techniques, but also some newer methods. In particular, we introduce a Borsuk-Ulam theorem for flag manifolds, which we believe to be of independent interest .
After revision:  This question was already studied in the plane, but in higher dimensions the only known results are for the case where k is an odd prime. We extend these results to a larger family of values of k. We further present a new result for k=2, which generalizes to cones. The second question considers bisections with double wedges or, equivalently, Ham-Sandwich cuts after projective transformations. Here we prove that given d families of d+1 point sets each, there is always a projective transformation such that after the transformation, each family has a Ham-Sandwich cut. We further prove a result on partitions with parallel hyperplanes after a projective transformation .