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Taxicab norm or Manhattan norm

‖ x ‖ 1:= ∑ i = 1 n | x i |. {\displaystyle \|{\boldsymbol {x}}\|_{1}:=\sum _{i=1}^{n}\left|x_{i}\right|.} The name relates to the distance a taxi has to drive in a rectangular street grid (like that of the New York borough of Manhattan) to get from the origin to the point x. {\displaystyle x.}

The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope, which has dimension equal to the dimension of the vector space minus 1.The Taxicab norm is also called the ℓ 1 {\displaystyle \ell ^{1}} norm. The distance derived from this norm is called the Manhattan distance or ℓ 1 {\displaystyle \ell ^{1}} distance.

The 1-norm is simply the sum of the absolute values of the columns.

In contrast, ∑ i = 1 n x i {\displaystyle \sum _{i=1}^{n}x_{i}} is not a norm because it may yield negative results.