Revistes Catalanes amb Accés Obert (RACO)

English summaries


Resum


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 the so-called John
ellipsoid 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.

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