English summaries - , (Butlletí de la Societat Catalana de Matemàtiques, 2017)
In this paper we present some notions and classical results from convex geometrywhich have found numerous applications. We concentrate on threefamilies of convex bodies: ellipsoids, centrally symmetric convex bodies andzonoids, and describe some of their applications in geometry. For instance, weprove Minkowski's first theorem on the geometry of numbers, the existenceof an ellipsoid of maximal volume inside a convex body Âthe so-called JohnellipsoidÂ and study Shephard's problem, which asks if there are pairs ofbodies one with a smaller volume than the other, but with larger projections.The centrally symmetric bodies and the zonoids are also described as the rangeof certain operators: the difference and projection operators. At the beginningof this paper we present the basic notions of convex geometry that will be usedthroughout and take a brief look at the combinatorial geometry, presentingHelly's theorem and some of its consequences.