coverdensity

Coverdensity is a term used in mathematics and applied sciences to describe how densely a region is covered by a collection of sets. In two dimensions, let R be a region with finite area and {C_i} a finite or countable family of measurable subsets that aim to cover R. The coverdensity D of the cover on R is defined as D = sum_i area(C_i ∩ R) divided by area(R). This definition extends to higher dimensions by replacing area with volume or measure. If all covering sets lie within R, the formula reduces to D = sum_i area(C_i)/area(R).

Interpretation: If the union of coverings equals R and there is no overlap, D = 1. If the union is R but there is overlap, D > 1. If the union does not cover all of R, D < 1. Thus D measures both coverage and redundancy: higher values reflect more overlap or redundancy, while lower values reflect gaps in coverage.

Applications include assessing coating efficiency in materials science, evaluating tilings and coverings in geometry, and estimating sensing or service coverage in spatial networks. For example, covering a unit square by two overlapping unit squares yields D = 2, while disjoint tiles that tile the region give D = 1. In GIS, D serves as a simple quantitative measure of how much of a region is serviced or observed by a set of sensors or facilities.

See also: coverage problem, set cover, tiling, geometric measure theory.

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