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

On relative isodiametric inequalities

We consider subdivisions of convex bodies G in two subsets E and G\E. We obtain several inequalities comparing the relative volume 1) with the minimum relative diameter and 2) with the maximum relative diameter. In the second case we obtain the best upper estimate only for subdivisions determined by straight lines in planar sets

On fencing problems

Fencing problems deal with the bisection of a convex body in a way that some geometric measures are optimized. We study bisections of planar bounded convex sets by straight line cuts and also bisections by hyperplane cuts for convex bodies in higher dimensions