Existence of least-perimeter partitions

Least - Perimeter Partitions of the Sphere

We consider generalizations of the honeycomb problem to the sphere S and seek the perimeter-minimizing partition into n regions of equal area. We provide a new proof of Masters’ result that three great semicircles meeting at the poles at 120 degrees minimize perimeter among partitions into three equal areas. We also treat the case of four equal areas, and we prove under various hypotheses that ...

Minimum Perimeter-Sum Partitions in the Plane

Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P1 and P2 such that the sum of the perimeters of ch(P1) and ch(P2) is minimized, where ch(Pi) denotes the convex hull of Pi. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n2) time algorithm. Despite considerable progress on related problems, no subquadratic time algo...

A note on non lower semicontinuous perimeter functionals on partitions

We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in R. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity.

Existence of Abelian Group Code Partitions

Duadic codes were introduced by Leon et al. [5] as cyclic codes generalizing quadratic residue codes. Brualdi and Pless generalized them to polyadic cyclic codes [2] and Rushanan did so to duadic Abelian group codes [9]. Theorems concerning the existence of these codes in terms of field and group restrictions have been proved during this development, beginning with the one of Smid for cyclic du...

Existence and nonexistence of G � least energy