Trisections of a 3-rotationally symmetric planar convex body minimizing the maximum relative diameter

In this work we study the fencing problem consisting of finnding a trisection of a 3-rotationally symmetric planar convex body which minimizes the maximum relative diameter. We prove that an optimal solution is given by the so-called standard trisection. We also determine the optimal set giving the minimum value for this functional and study the corresponding universal lower bound.Comment: Preliminary version, 20 pages, 15 figure

A new bound on the Morse index of constant mean curvature tori of revolution in S3

In this work we give a new lower bound on the Morse index for constant mean curvature tori of revolution immersed in the three-sphere S3, by computing some explicit negative eigenvalues for the corresponding Jacobi operator.Ministerio de Educación y Ciencia MTM2007-6191

The maximum relative diameter for multi-rotationally symmetric planar convex bodies

In this work we study the maximum relative diameter functional dM in the class of multi-rotationally symmetric planar convex bodies. A given set C of this class is k-rotationally symmetric for k 2 {k1, . . . , kn} N, and so it is natural to consider the standard ki-partition Pki associated to C (which is a minimizing ki-partition for dM when ki > 3) and the corresponding value dM(Pki ). We establish the relation among these values, characterizing the particular sets for which all these values coincide.Ministerio de Economía y Competitividad MTM2013-48371-C2-1-PJunta de Andalucía FQM-32