Lengthssuch

Lengthssuch is a nonstandard term used in some mathematical discussions to denote a family of length-like invariants assigned to curves, paths, or more general 1-dimensional objects. It is not a universally standardized concept, but rather a placeholder for various generalized notions of length proposed by different authors. The term is typically employed to compare or contrast different ways of measuring the size of a path beyond classical arc length.

In general, a lengthssuch L assigns a nonnegative real value to each rectifiable curve γ in a space X, with the specific axioms depending on the chosen framework. Common features across treatments include non-negativity, L(γ) = 0 only for constant curves, invariance under reparameterization, and additivity under concatenation of compatible paths. Some approaches also require monotonicity with respect to subpaths or compatibility with a given metric or measure. Because there is no single universal definition, different authors may impose additional properties such as lower semicontinuity, homogeneity under scaling, or compatibility with subspaces.

Examples often cited under the umbrella of lengthssuch include the standard Euclidean arc length, lengths induced by weighted edge graphs, and various p-norm or sub-Riemannian generalizations. In fractal or non-smooth spaces, lengthssuch notions may diverge from classical arc length, yielding alternative measurements that reflect local geometry.

Usage and history: the term arises mainly in informal notes and seminar discussions as a way to discuss length concepts without committing to a single formalism. As such, references and precise axioms vary by source, and readers should consult the specific framework being discussed in a given work. See also length, arc length, metric, and geometric measure theory.