Equipartition of mass distributions by hyperplanes

We consider the problem of determining the smallest dimension d = Delta(j, k) such that, for any j mass distributions in R(d), there are k hyperplanes so that each orthant contains a fraction 1/2(k) of each of the masses. The case Delta(1, 2) = 2 is very well known. The case k = 1 is answered by the ham-sandwich theorem with Delta(j, 1) = j. By using mass distributions on the moment curve the lower bound Delta(j, k) greater than or equal to j(2(k) - 1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Delta(j, k) less than or equal to j2(k-1). We are able to prove that Delta(j, k) = [j(2(k) - 1)/k] for a few pairs (j, k) ((j, 2) for j = 3 and j = 2(n) with n greater than or equal to 0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Delta(1, 4) (the only case for j = 1 in which it is not known whether Delta(1, k) = k); unfortunately the approach fails to give an answer in this case (but we can show Delta(1, 4) less than or equal to 5).