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In this paper we construct a family of spin Lie groups G with an outer automorphism of order three (triality automorphism) and we describe the subgroups of fixed points for this kind of automorphisms. We will take advantage of this work to study the action of the group of outer automorphisms of G on the moduli space of principal G-bundles and describe the subvariety of fixed points in M(G) for the action of the outer automorphism of order three of G. Finally, we further study the case of Spin(8, C). Resumen En este artículo construimos una familia de grupos de Lie espinoriales G dotados de un automorfismo externo de orden tres (trialidad) y describimos los subgrupos de puntos fijos para esta clase de automorfismos. Usaremos esto para estudiar la acción del grupo de automorfismos externos de G en el espacio de moduli de G-fibrados principales y describir la subvariedad de puntos fijos en M(G) para la acción del automorfismo externo de orden tres de G. Finalmente, profundizaremos en el estudio del caso Spin(8, C).

Problems related to the random approximation of convex bodies fall into the field of integral geometry and geometric probabilities. The aim of this paper is to give a survey of known results about the stochastic model that has received special attention in the literature and that can be described as follows: Let K be a d-dimensional convex body in Eucliden space Rd, d = 2. Denote by Hn the convex hull of n independent random points X1, ..., Xn distributed identically and uniformly in the interior of K. If f is a random variable on d-dimensional polytopes on Rd, we define the random variable fn by: fn = f (conv {X1, ..., Xn}), where conv denotes the convex hull. Typical random variables studied in the literature are numbers of vertices and facets, volume, surface area and mean width. Our main interest concerns the study of the mathematical expectation E(fn) of fn. Some further stochastic models and other problems related to random points studied in the literature will be presented.