CONVEX DOMAINS

Author: MARTÍN JIMÉNEZ PEDRO.
Year: 2002.
University: EXTREMADURA [www.unex.es].
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: FACULTAD DE CIENCIAS.

Summary: The ellipsoids are the only bodies convex in which all sections flat are elipses.Asimismo, are the only bodies with convex center, which are symmetrical with respect to any plane passing through its centro.En this memory weaken the hypothesis of these characterizations restricting the number of sections and symmetries. Chapter 1 contains general definitions are handled along the memory, as well as a brief historical overview, which contains the main results related to the topic. Be a body B convex space akin E3. Chapter 2 is devoted to studying how many beams are required for planes that B has to be an ellipsoid, knowing that the levels of these sections you are elipses.Se shows, for example, that if rys are two parallel lines, one of which passes inside B, and all sections with flat or as containing ar are elliptical, then B must be a elipsoide.Se obtained similar results when rys are cut or are drying, and when considering beam planes paralelos.El chapter complete with a collection of examples and counterexamples. In Chapter 3 shows that if B is symmetrical with respect to three levels and there is some relationship between the plans and directions of symmetry, then B must be a elipsoide.En another section of this chapter shows that the relationships that the regions the Voronoi the plane are convex if and only if the distance is defined euclídea. In chapter 4 covers dimension greater than three results of the previous chapters. The report ends with an extensive bibliography on the subject.

Author: MIORI CINZIA.
Year: 2006.
University: ALICANTE [www.ua.es].
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: UNIVERSIDAD DE ALICANTE.

Summary: Those problems are studied in those looking for an optimal solution to the problem of dividing a convex set and compact in two subsets of equal volume (area) to optimize certain features geometric subsets of genetic dice. Numerous results (Eggleston, Bokowski, Cianchi, â |) if it seeks to minimize the perimeter of the "near" that divides the given set (also called on the perimeter), and also several problems on open heights give the best overall estimate , some of them raised by Santaló. In particular have been identified sets extremales and has sought global bounds on the best possible subdivision. They are considering both the simplest subdivision by straight (in the case plan) or hiperplanos the branch more general curves (in the case plan). We have also considered the study to other geometric figures: in the case level have been studied further geometric properties that relate quantities classic with the length of the maximum and minimum rope that divides the set given in two regions of the same area. The work plan has been worked with joint plans for subdivisions and straight, looking first obtain the best branch for individual examples of joint, then set centrally symmetrical and finally set convex general have been studied in this case the properties goemétricas the sets obtained and its possible uniqueness; then have sought the best overall heights.

Author: HERRERO PIÑEYRO PEDRO JOSÉ.
Year: 2006.
University: MURCIA [www.um.es].
Place of defense: FACULTAD DE MATEMÁTICAS.
Place of preparation: FACULTAD DE MATEMÁTICAS.

Summary: The objective of this work has been the study of the ring at least a set convex plane, as well as its relationship with other geometric quantities, this leads to obtaining the best possible inequalities among the measures considered. Given a convex body K (convex set and compact), defines a ring centered cy minor radii r R as the closed set consisting of dots between the fields (concentric) center cy rays ry R. The convexity also implies the existence of a single ring with different radio Rr floor, it is called the ring less than whole. Following the work initiated by Bonnesen, Favard and others, we have looked at, first, the relationship between the ring convex body of a minimum level with each of the six classic geometric figures: the area, perimeter, diameter, minimum width, and circumradius inradius. In more precise, we determine all possible bounds (upper and lower) for such measures if we assume that the ring is fixed minimum. Note that if the circuncírculo and incírculo a convex body K are concentric, of course they determine the minimum set ring (for example, in the square). But why does not always have such a thing happen: in fact, may be any of the other possibilities. This has motivated our interest for studying the relationship between the minimum ring of a set, his circuncírculo and incírculo. So, after demonstrating various properties that relate these geometric objects, all inequalities best determine which set what figures that maximize or minimize each of the previous measures when its minimum ring and either circumradius his or her inradius, they are fixed. It solved all possible cases, which closes the problem completely.