• Zonoid
• Lift Zonoid
• Depth Statistics
• Gini Volume
• Brightness Function
• Convex Geometry
• Production Theory

Many problems in the social and system sciences are naturally multivariate and cannot be easily represented with a continuous or parametric approach.
An example is the economical production theory, that is, the theory that studies and represents the determinant factors driving production process dynamics. An industry is defined as a set of firms operating within the same sector and we can think about firm productivity as the ``ability" to turn inputs into outputs.
The classic approach in production theory is based on a number of assumptions regarding firms behaviour and firm production possibilities, in particular the profit maximization and cost minimization assumption. Following these assumptions, an ad hoc parametrized family of production functions is introduced to assess firm productivity and efficiency and to estimate a number of economical indices. Such production functions satisfy, in addition, certain topological properties such as convexity and continuity, thus implying that firms with similar technologies will adopt analogous production techniques or, equivalently, firms tend to be homogeneous.
Despite these assumptions, a growing availability of longitudinal microdata at firm-level has evidenced the fundamental role of heterogeneity in all relevant aspects regarding firms production activity, thus suggesting a switch from a continuous/parametric approach (which seems to be inadequate in presence of wide asymmetries) to a discrete/nonparametric point of view. Here geometry and geometric measure theory come into help.
To evidence the fragilities of the classic theory, in 1981 Hildenbrand adopted a geometric representation of the empirical distribution of the industry, that is, the zonoid representation. Geometrically, a zonoid is a centrally symmetric, compact, convex set of the euclidean space which is induced by a Borel measure with finite expectation. In particular, the zonoid induced by the empirical distribution of a given industry is a convex polytope which is called a zonotope. Zonotopes can also be written as a sum of line segments, in addition they are dense in the class of zonoids with respect to the topology induced by the Hausdorff metric.
More recently, in a 2016 paper, Dosi, Grazzi, Marengo and Settepanella adopted Hildenbrand's construction to assess the rate of productivity and technological change of a given industry both on the microeconomical point of view (i.e. firm-level productivity) and on the macroeconomical point of view (i.e. aggregate productivity). Moreover a measure of heterogeneity of the industry, called the Gini volume, is introduced. The above approach relies entirely on the geometry of the zonotope induced by the empirical distribution of the industry and it is highly nonparametric, on the other hand the Gini volume can also be seen as a measure of dispersion of the emprical distribution.
The aim of the following thesis is to investigate the duality, provided by the zonoid representation, between the geometry of convex bodies and the geometric measure theory of Borel measures with finite first moment. In this respect, new methods of nonparametric statistics may derive from new geometrical insights.