perimeterlike

Perimeterlike is a term used in geometry and related disciplines to describe quantities that play a role analogous to the perimeter of a figure—the measure of its boundary. A perimeterlike functional F defined on a class of sets is intended to capture boundary-length properties, often sharing two key features: monotonicity with respect to set inclusion (A ⊆ B implies F(A) ≤ F(B)) and additivity over disjoint unions (F(A ∪ B) = F(A) + F(B) when A and B are disjoint). In addition, such a measure usually scales with similarity transformations in ways similar to length: F(tA) = tF(A) for scalings.

In classical planar geometry, the standard perimeter P(A) is the canonical example of a perimeterlike quantity. In other contexts, especially in discrete or irregular settings, practitioners speak of perimeterlike measures such as boundary length in pixel grids or contour length in digital images. These discrete counterparts approximate the Euclidean perimeter but may depend on the chosen discretization.

Limitations: Some sets have finite area but infinite classical perimeter, as in fractal boundaries like the Koch snowflake, which motivates the need for generalized or alternative perimeterlike functionals.

Applications: Perimeterlike measures appear in shape analysis, image processing, and materials science, where boundary properties determine behavior such as diffusion, strength, or recognition features.

Etymology: The term perimeterlike is a descriptive, nonstandard label indicating analogy to boundary length rather than a formal definition.