Summary: We consider the problem of determining the smallest dimension \(d = \Delta (j, k)\) such that, for any \(j\) mass distribution in \(\mathbb{R}^d\), 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 \(\Delta (j,k) \geq j (2^k - 1)/k\) is obtained. We believe this is a tight bound. However, the only general upper bound that we know is \(\Delta (j,k) \leq j2^{k - 1}\). We are able to prove that \(\Delta (j,k) = \lceil j(2^k - 1)/k \rceil \) for a few pairs \((j,k)\) \(((j,2)\) for \(j = 3\) and \(j = 2^n\) with \(n \geq 0\), 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 \(\Delta (1,4) \leq 5)\).