English summaries - , (Butlletí de la Societat Catalana de Matemàtiques, 2017)
In this paper we present some notions and classical results from convex geometry which have found numerous applications. We concentrate on three families of convex bodies: ellipsoids, centrally symmetric convex bodies and zonoids, and describe some of their applications in geometry. For instance, we prove Minkowski’s first theorem on the geometry of numbers, the existence of an ellipsoid of maximal volume inside a convex body Âthe so-called John ellipsoidÂ and study Shephard’s problem, which asks if there are pairs of bodies one with a smaller volume than the other, but with larger projections. The centrally symmetric bodies and the zonoids are also described as the range of certain operators: the difference and projection operators. At the beginning of this paper we present the basic notions of convex geometry that will be used throughout and take a brief look at the combinatorial geometry, presenting Helly’s theorem and some of its consequences.