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

Minimum number of different distances defined by a finite number of points

We study the minimum number of different distances defined by a finite number of points in the following cases: a) we consider metrics different from the euclidean distance in the plane, b) we consider the euclidean distance but restricted to subsets of the plane of special interest, c) we consider other topological surfaces: the cylinder and the flat torus. All these results extend those obtained by Erdös and other mathematicians for the euclidean distance in the plane

Bisections of centrally symmetric planar convex bodies minimizing the maximum relative diameter

In this paper we study the bisections of a centrally symmetric planar convex body which minimize the maximum relative diameter functional. We give necessary and sufficient conditions for being a minimizing bisection, as well as analyzing the behavior of the so-called standard bisection.Ministerio de Economía e Innovación MTM2013-48371-C2-1-PJunta de Andalucía FQM-32

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

Subdivisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter

In this work we study subdivisions of k-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called k-partitions, consisting of k curves meeting in an interior vertex, we prove that the so-called standard k-partition (given by k equiangular inradius segments) is minimizing for any k 2 N, k > 3. For general subdivisions, we show that the previous result only holds for k 6 6. We also study the optimal set for this problem, obtaining that for each k 2 N, k > 3, it consists of the intersection of the unit circle with the corresponding regular k-gon of certain area. Finally, we also discuss the problem for planar convex sets and large values of k, and conjecture the optimal k-subdivision in this case.Ministerio de Educación y Ciencia MTM2010-21206-C02-01Ministerio de Economía e Innovación MTM2013-48371-C2-1-PJunta de Andalucía FQM-325Junta de Andalucía P09-FQM-508

Seguimiento Grado en Matemáticas. Curso 13-14

El objetivo principal de esta red ha sido la coordinación y seguimiento de los cursos correspondientes al Grado en Matemáticas que se ha implantado en su totalidad en el presente curso académico en la Facultad de Ciencias de la Universidad de Alicante y se engloba dentro del proceso general del seguimiento de todos los títulos de la Facultad de Ciencias. La red está coordinada por la coordinadora del Grado en Matemáticas y formada por los coordinadores de cada uno de los semestres. Se pretende evidenciar los progresos del título en el desarrollo del Sistema de Garantía Interno de Calidad (SGIC), con el fin de detectar las posibles deficiencias en el proceso de implantación del grado y contribuir a sus posibles mejoras elaborando propuestas de acciones para mejorar su diseño y desarrollo

Bisections of Centrally Symmetric Planar Convex Bodies Minimizing the Maximum Relative Diameter

In this paper, we study the bisections of a centrally symmetric planar convex body which minimize the maximum relative diameter functional. We give a necessary and a sufficient condition for a minimizing bisection, as well as analyze the behavior of the so-called standard bisection.Antonio Cañete is partially supported by the MICINN projects MTM2013-48371-C2-1-P and MTM2017-84851-C2-1-P, and by Junta de Andalucía grant FQM-325 (Consejería de Economía, Innovación, Ciencia y Empleo). Salvador Segura Gomis is partially supported by MINECO/FEDER project MTM2015-65430-P and “Programa de Ayudas a Grupos de Excelencia de la Región de Murcia”, Fundación Séneca, 19901/GERM/15

Two Problems Concerning the Area-perimeter Ratio of Lattice-point-free Regions in the Plane

. We give a generalization of Bender's area-perimeter relation for plane lattice-point-free convex regions to simply connected regions, thus we solve a problem posed by M. Silver [10]. Further the result is used for a lattice version of the Dido problem. MSC 1991: 52C05, 11H06 1. Introduction and Results For a plane lattice-point-free convex region with area A and perimeter P Bender [1] proved the relation A P ! 1 2 : This result has been generalized in different ways. There are complete solutions for the generalization to higher dimensions [4] and to arbitrary lattices for d = 2 [9]. Further there are results for convex bodies containing a certain number of lattice points ([2],[7],[8]). The work has been partially supported by Consejeria de Cultura y Educacion de la C.A.R.M. PB 94-10 and by the DGICYT (Spain) Grant no. PB 91-0324. 0138-4821/93 $ 2.50 c fl 1996 Heldermann Verlag, Berlin 2 U. Schnell, S. Segura Gomis: Two Problems Concerning the Area-perimeter Ratio ... Here..

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