Uneven splitting of ham sandwiches (Q972599)

From the abstract: Let \(\mu_1,\dots,\mu_n\) be continuous probability measures on \(\mathbb R^n\) and \(\alpha_1,\dots,\alpha_n \in [0,1]\). When does there exist an oriented hyperplane \(H\) such that the positive half-space \(H^+\) has \(\mu_i (H^+)=\alpha_i\) for all \(i\in[n]\)? It is well known that such a hyperplane does not exist in general. The famous Ham Sandwich Theorem states that if \(\alpha_i=\frac{1}{2}\) for all \(i\), then such a hyperplane always exists. In this paper, the author gives sufficient criteria for the existence of \(H\) for general \(\alpha_i \in [0,1]\). Let \(f_1,\dots,f_n :S^{n-1}\to \mathbb R^n\) denote auxiliary functions with the property that for all \(i\), the unique hyperplane \(H_i\) with normal \(v\) that contains the point \(f _i(v)\) has \(\mu_i (H_i^+)=\alpha_i\). His main result is that if \(\text{Im } f_1,\dots,\text{Im } f_n\) are bounded and can be separated by hyperplanes, then there exists a hyperplane \(H\) with \(\mu_i (H^+)=\alpha_i\) for all \(i\). This gives rise to several corollaries; for instance, if the supports of \(\mu_1,\dots,\mu_n\) are bounded and can be separated by hyperplanes, then \(H\) exists for any choice of \(\alpha_1,\dots,\alpha_n \in[0,1]\). He also obtains results that can be applied if the supports of \(\mu_1,\dots,\mu_n\) overlap.