Choosing the center of star-shaped set-valued data compatible with measure-preserving arithmetic

Set-valued data has traditionally been represented by considering non-empty compact and convex subsets of $R^d$ with the usual Minkowski addition. An alternative and flexible setting that admits a functional representation are the star-shaped sets. A framework based on a center-radial characterization has been introduced to treat these sets from a statistical point of view. The arithmetic is defined directionally, which is more natural for representing imprecision propagation in higher dimensions. Nevertheless, the problem of determining a center for star-shaped sets coherent with the arithmetic and sound for statistical purposes has not been fully addressed yet. The aim is to advance on the directional characterization for star-shaped sets by considering a measure-preserving arithmetic together with a center selection fully compatible with this arithmetic. The practicability of the new framework will be illustrated using a classical dataset in set-valued statistics.