Before revision:  The first one is , whether we can bisect more than d masses, if we allow for bisections with more general objects such as cones, wedges or double wedges. We answer this question in the affirmative by showing that with all of these objects, we can simultaneously bisect d+1 masses. For double wedges, we prove a stronger statement, namely that d families of d+1 masses each can each by simultaneously bisected by some double wedge such that all double wedges have one hyperplane in common. The second question is, how many masses we can simultaneously equipartition with a k-fan, that is, k half-hyperplanes in R^d, emanating from a common (d-2)-dimensional apex.
After revision:  The first one is how many masses we can simultaneously equipartition with a k-fan, that is, k half-hyperplanes in R^d, emanating from a common (d-2)-dimensional apex.