Measures in Convex Geometry
By convex geometry we understand here the geometry of convex bodies in Euclidean space. In this field, measure theory enters naturally and is useful under several different aspects. First, like in many other fields, measures are employed to quantify the smallness of certain exceptional sets. In our first chapter, we give examples showing how Hausdorff measures of different dimensions are appropriate tools for describing sets of singular points or directions related to the boundary structure of convex bodies. In the second chapter we treat measures that are designed to refflect the local behaviour of convex bodies in a simi¬lar way as curvatures are used in differential geometry. The third connection between convex geometry and measure theory that we want to explain is of an entirely different nature. Here we treat a special class of convex bodies, the zonoids, which can be defined in terms of measures, and we show by an example from stochastic geometry how they are related to other fields. The second of these topics will be treated in greater detail than the other two. Naturally, some facts from the geometry of convex bodies will have to be used without proof. The fundamental notions will be explained and are easy to understand, due to their intuitive character. As a reference where proofs can be found, we mention the book .
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
29 (1998) s.
Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Rolf Schneider, "Measures in Convex Geometry", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 29 (1998) suppl., pp. 215-265.