Hypermedian

Hypermedian is a term used in several mathematical contexts to denote a generalized central tendency measure extending the univariate median into higher dimensions or more complex structures. There is no single, universally accepted definition. In data analysis, hypermedian is sometimes used to refer to the coordinate-wise (componentwise) median: for a set of d-dimensional points, the hypermedian is the point formed by taking the median of each coordinate across the data. This gives a robust center that lies within the axis-aligned bounding box of the sample but does not generally minimize the sum of distances.

In other literature, hypermedian is used to denote the geometric median (also called the spatial median or Fermat-Weber point). This multivariate generalization minimizes the sum of Euclidean distances to the sample points and is a natural extension of the univariate median in the metric sense. Unlike the coordinate-wise median, it takes all coordinates jointly into account and may be unique under mild conditions. Computation often uses iterative methods such as Weiszfeld's algorithm.

In hypergraph and network contexts, hypermedian can refer to a center of a set of vertices that minimizes some distance-based objective over hyperedges or hyperpaths. Definitions vary with the chosen distance or hypergraph metric, so the term is largely context-dependent.

Because of varied usage, many authors prefer to specify the exact objective (coordinate-wise median, geometric median, spatial median, or a hypergraph center) rather than the umbrella term hypermedian. The term remains informal and is not standardized across disciplines.