Stationary partitioning of convex bodies
Let \({\mathcal B}\) be a convex body in \({\mathbb{R}}^ 3\) of volume \(m_ 3({\mathcal B})<\infty\). A disc-type surface \({\mathcal S}\) is spanned across the interior of \({\mathcal B}\) dividing this interior in two simply-connected cells \({\mathcal B}'\) and \({\mathcal B}''\) of volumes \(m_ 3({\mathcal B}')=\sigma m_ 3({\mathcal B}),\quad m_ 3({\mathcal B}'')=(1-\sigma)m_ 3({\mathcal B})\) \((0<\sigma <1)\). The fraction \(\sigma\) is considered as fixed and the problem is to find \({\mathcal S}\) with the smallest possible area or with stationary area. Both conditions imply that the first variation of the area of \({\mathcal S}\) vanishes. A surface with these properties is called stationary. Theorem: For a ball \({\mathcal B}\), all stationary surfaces (which are orientable, simply connected and have Euler characteristic one) are plane discs through the center \((\sigma =1/2)\) and spherical caps orthogonal to \(\partial {\mathcal B}\) (\(\sigma\) \(\neq 1/2).\) Other interesting examples are considered and noteworthy open questions are stated.
scientific article; zbMATH DE number 3913146
scientific article