The parametric partitioning problem (Q2781067)

The topic of this thesis is the partitioning problem. Here, for a given compact body \(K\subset\mathbb{R}^3\) one is looking for a separating surface \(S\) dividing the given volume \(V(K)\) into two prescribed volumes such that the area of \(S\) becomes minimal. In his approach the author considers \(S\) as a parametrized surface given by a mapping \(X: B\to\mathbb{R}^3\) with \(X(\partial B)\subset\partial K\), \(B\subset\mathbb{R}^2\) being the unit disc. The two main results are an optimal a-priori-estimate for the mean curvature of a solution \(\Sigma\) to the partitioning problem (Satz 2.9), and an existence result (Satz 3.30). The advantage of the parametric approach compared to the one using methods from geometric measure theory is the ability to control the topological type of the solution \(\Sigma\). In fact, Bürger's existence theorem yields a solution which consists of finitely many images of the unit disc. A conjecture by Ros, namely that the solution for a convex body \(K\) consists of only such disc, still remains open.