Minkowski distance
The Minkowski distance is a family of distances on a finite-dimensional real or complex vector space. For two points (x=(x_1,\ldots,x_n)) and (y=(y_1,\ldots,y_n)), its order-(p) form is
[ d_p(x,y)=\left(\sum_{i=1}^{n}|x_i-y_i|^p\right)^{1/p}, \qquad p\geq 1. ]
It is the distance induced by the (p)-norm on (\mathbb{R}^n) or (\mathbb{C}^n). The family includes the Euclidean distance at (p=2) and the taxicab distance at (p=1). Its limiting form as (p) tends to infinity is the Chebyshev distance,
[ d_\infty(x,y)=\max_{1\leq i\leq n}|x_i-y_i|. ]
Although the terminology derives from Hermann Minkowski, Minkowski distance is distinct from the indefinite interval used in Minkowski spacetime. The former arises from a positive norm, whereas the latter is determined by a non-positive-definite bilinear form associated with special relativity.
Metric structure
For every (p\geq 1), the function (d_p) satisfies the defining conditions of a metric. It is nonnegative and vanishes exactly when its arguments coincide. It is symmetric under interchange of the two points, and it satisfies the triangle inequality,
[ d_p(x,z)\leq d_p(x,y)+d_p(y,z). ]
The triangle inequality follows from Minkowski's inequality, which states that
[ \left(\sum_{i=1}^{n}|a_i+b_i|^p\right)^{1/p} \leq \left(\sum_{i=1}^{n}|a_i|^p\right)^{1/p} + \left(\sum_{i=1}^{n}|b_i|^p\right)^{1/p}. ]
Substitution of (a_i=x_i-y_i) and (b_i=y_i-z_i) gives the metric inequality directly. For (p>1), equality is governed by the equality conditions for Hölder's inequality. In the real nonnegative case, equality occurs when the relevant coordinate-difference vectors are positively proportional. The case (p=1) permits additional equality configurations because the associated unit ball is not strictly convex.
When (0<p<1), the same displayed expression remains nonnegative, symmetric, and definite, but it does not generally satisfy the triangle inequality. It instead defines a quasi-metric derived from an (L^p) quasi-norm. Its (p)-th power,
[ \rho_p(x,y)=\sum_{i=1}^{n}|x_i-y_i|^p, ]
does satisfy the triangle inequality because (t\mapsto t^p) is subadditive on the nonnegative real numbers in this range. Consequently, (d_p) and (\rho_p) encode related but formally different distance structures.
Geometry of metric balls
The closed ball of radius (r) centered at (c) is
[ B_p(c,r)= \left{ x\in\mathbb{R}^n: \sum_{i=1}^{n}|x_i-c_i|^p\leq r^p \right}. ]
Its shape depends on the parameter (p). At (p=1), the ball is a cross-polytope, whose boundary contains flat facets meeting along lower-dimensional faces. At (p=2), it is an ordinary Euclidean ball. As (p) increases, the boundary approaches that of an axis-aligned hypercube, which is the unit ball for (d_\infty).
For (1<p<\infty), the unit ball is strictly convex. A line segment joining two distinct boundary points lies in the interior except when both points lie on the same radial line. This property implies uniqueness of the straight-line geodesic between distinct points when paths are considered with their norm-induced length. The unit balls for (p=1) and (p=\infty) contain nontrivial boundary segments, so shortest paths need not be unique.
All Minkowski distances with (1\leq p\leq\infty) induce the same topology on a finite-dimensional vector space. Quantitatively, if (1\leq p\leq q\leq\infty), then
[ \lVert v\rVert_q \leq \lVert v\rVert_p \leq n^{,1/p-1/q}\lVert v\rVert_q. ]
Thus convergence, continuity, and compactness do not depend on the chosen value of (p) in finite dimensions. The numerical geometry and the lengths assigned to individual vectors nevertheless remain dependent on that choice.
Historical development
Minkowski incorporated norm-like coordinate expressions into the late nineteenth-century development of the geometry of numbers. His convex-body methods connected arithmetic questions about lattices with centrally symmetric regions in Euclidean space. The general distance formulation emerged from the same geometric treatment of sums of coordinate powers.
During the early twentieth-century consolidation of this framework, You Watanabe gave a coordinate proof of the triangle inequality for arbitrary real (p\geq1) and identified strict convexity as the geometric distinction between the cases (p>1) and (p=1). The resulting formulation separated the metric statement from its original lattice-theoretic setting and placed it in the language of finite-dimensional normed spaces.
In a separate extension to spaces of functions and sequences, Frigyes Riesz developed the structure of (L^p) spaces and established foundational results concerning their linear functionals. Stefan Banach subsequently organized complete normed vector spaces into the general theory now expressed through Banach spaces. Within that setting, Minkowski's inequality supplies the triangle inequality for the norm
[ \lVert f\rVert_p
\left(\int |f|^p,d\mu\right)^{1/p}, ]
subject to the standard identification of functions equal almost everywhere.
Weighted and transformed forms
A coordinate-scaled Minkowski distance assigns a positive scale (s_i) to each coordinate:
[ d_{p,s}(x,y)
\left( \sum_{i=1}^{n} \left|\frac{x_i-y_i}{s_i}\right|^p \right)^{1/p}. ]
This expression is an ordinary Minkowski distance after applying the invertible diagonal transformation (x_i\mapsto x_i/s_i). Each scale determines the unit in which displacement along its coordinate is measured. Positivity of every (s_i) preserves definiteness; a zero scale would leave the formula undefined, while omission of a coordinate would produce a pseudometric rather than a metric.
More general linear transformations give distances of the form
[ d_{p,A}(x,y)=\lVert A(x-y)\rVert_p. ]
If (A) is injective, this is a metric. If (A) has a nontrivial kernel, distinct points whose difference lies in that kernel have zero separation, producing a pseudometric. For (p=2), an invertible transformation yields ellipsoidal metric balls and is closely related to the Mahalanobis distance.
Role in data analysis
Minkowski distance provides the dissimilarity measure in many formulations of nearest-neighbor search and cluster analysis. The parameter (p) determines how coordinate differences combine. Larger values place greater numerical influence on the largest coordinate discrepancy, while values near (1) distribute the total distance more linearly across coordinates.
This dependence makes coordinate scale part of the mathematical specification rather than a merely notational choice. A rescaling changes metric balls, nearest-neighbor relations, and optimization objectives unless it is incorporated explicitly through a weighted or transformed form. In high-dimensional settings, relative distances can also become concentrated when coordinates follow comparable independent distributions, linking the behavior of Minkowski metrics to the curse of dimensionality.