R. v. Mrs Utam Singh

Efficient algorithm for optimizing spectral partitions

“…In [25] it is mentioned that these polyhedra are dense in the class of 3D constant width bodies using arguments based on [26]. The density of Meissner polyhedra in the class of three dimensional constant width bodies was revisited in [12] where a detailed description is given regarding the construction of these bodies.…”

“…Moreover, a detailed description of the computation of the volume and surface area for the Meissner tetrahedra is given. The surface area and the volume of Meissner polyhedra is computed in [13]. In this paper an alternate computation is proposed, resulting in a simple formula depending only on the lengths of the pairs of dual edges.…”

“…the number of diametric pairs of X is maximal. In [12,Theorem 3.4] is recalled that a set is extremal if and only if there are 2m − 2 diametric pairs among points of X. Furthermore X is the set of vertices of B(X), i.e.…”

“…The resulting bodies are called Meissner polyhedra, since they generalize the Meissner tetrahedron. More details regarding these bodies, including their connections with extremal sets of diameter one and their density in the class of shapes of constant width are presented in [12]. It turns out that Meissner polyhedra give a natural context for studying Conjecture 1, since the Meissner tetrahedron naturally belongs to all classes of Meissner polyhedra which have an upper bound on the number of vertices.…”

“…These graphs are used for creating some of the illustrations in this paper. Recently, in [13] the surface area and volume of Reuleaux and Meissner polyhedra are computed, giving new tools for attacking Conjecture 1.…”