Inklusivvolumen

Inklusivvolumen is a term used in some German-speaking technical literature to denote the volume of a region including its boundary, i.e., the volume of the closure of a set in a given space. The term combines inklusiv (inclusive) and Volumen (volume). It is not universally standardized across disciplines, and its exact interpretation can vary.

In geometry and computational geometry, the inclusive volume of a region S ⊆ R^n often means the Lebesgue measure of the closure of S, denoted cl(S). If the boundary of S has measure zero (which is true for most well-behaved regions, such as polygons, polyhedra, and smooth manifolds), the inclusive volume equals the usual volume of S. In cases with irregular, fractal, or highly perforated boundaries, the inclusive volume may differ from the standard volume of S.

Applications include computer graphics, where inclusive volume may be used in voxel-based representations and collision detection, and in numerical integration where boundary effects are significant. It is sometimes contrasted with exclusive or interior volume, which counts only the points strictly inside S.

Because the term is not standardized, practitioners may prefer alternative phrases such as "closed volume" or "volume of the closure." See also volume, closure, measure, and boundary in geometric contexts.