sqrtlängd2p

sqrtlängd2p is a notation found in some discussions of metric geometry and data analysis, used to denote the square root of the distance between two points under a specified metric. The term combines the English prefix sqrt with the Swedish word längd (length) and the idea of a two-point distance (2p).

Formally, let d be a metric on a set X. For any two points p and q in X, sqrtlängd2p(p, q) is defined as the real number sqrt(d(p, q)). This transformation preserves nonnegativity and symmetry, and it maps zero distance to zero. Because sqrt is subadditive in the sense that sqrt(a + b) ≤ sqrt(a) + sqrt(b) for all nonnegative a, b, the function sqrtlängd2p also satisfies a form of the triangle inequality, making sqrtlängd2p a valid metric when d is a metric. In particular, if d is the Euclidean distance in R^n, then sqrtlängd2p(p, q) equals the Euclidean distance raised to the one-half power.

Common uses include distance-based algorithms where a different scale of distances is desirable, such as in clustering, multidimensional scaling, or kernel methods. Transforming distances by a square root can stabilize variance or emphasize small distances relative to larger ones.

Note on terminology: sqrtlängd2p is not a standard, widely adopted term in formal mathematical literature. It is a descriptive label that may appear in Swedish-language notes or informal discussions to describe the operation of taking a square root of a two-point distance under a chosen metric. See also distance, metric, and norm.