Before revision:  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 second question is, how many masses we can simultaneously partition with a k-fan, that is, k half-hyperplanes in R^d, emanating from a common (d-2)-dimensional apex.