Geometric median

The geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central tendency in higher dimensions. It is also known as the 1-median,spatial median,Euclidean minisum point, or Torricelli point.

The geometric median is an important estimator of location in statistics, where it is also known as the L₁ estimator. It is also a standard problem in facility location, where it models the problem of locating a facility to minimize the cost of transportation.

The special case of the problem for three points in the plane (that is, m = 3 and n = 2 in the definition below) is sometimes also known as Fermat's problem; it arises in the construction of minimal Steiner trees, and was originally posed as a problem by Pierre de Fermat and solved by Evangelista Torricelli. Its solution is now known as the Fermat point of the triangle formed by the three sample points. The geometric median may in turn be generalized to the problem of minimizing the sum of weighted distances, known as the Weber problem after Alfred Weber's discussion of the problem in his 1909 book on facility location. Some sources instead call Weber's problem the Fermat–Weber problem, but others use this name for the unweighted geometric median problem.

Wesolowsky (1993) provides a survey of the geometric median problem. See Fekete, Mitchell & Beurer (2005) for generalizations of the problem to non-discrete point sets.

Formally, for a given set of m points ${\displaystyle x_{1},x_{2},\dots ,x_{m}\,}$ with each ${\displaystyle x_{i}\in \mathbb {R} ^{n}}$, the geometric median is defined as

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