The “ham sandwich” theorem states that any $n$ finite Borel measures on $\mathbb{R}^{n}$ can be simultaneously bisected by a single hyperplane, provided each measure is absolutely continuous with respect to Lebesgue measure. In 1984, Cox & McKelvey showed that even for discontinuous measures, there exists a single hyperplane such that at most half of each measure lies on each side. In this talk, we consider the problem of minimizing the differences of the measures of the two open half-spaces determined by a chosen hyperplane, where the measures may be discontinuous. We show that if the dimension of $\mathbb{R}^{n}$ is much larger than the number of measures, then there exists a hyperplane that divides the measures more fairly than in Cox & McKelvey’s result.