Subpaving

In R¹ the boxes are line segments, in R² rectangles and in Rⁿ hyperrectangles.

Boxes present the advantage of being very easily manipulated by computers, as they form the heart of interval analysis.

Many interval algorithms naturally provide solutions that are regular subpavings.

[1] In computation, a well-known application of subpaving in R² is the Quadtree data structure.

In image tracing context and other applications is important to see X⁻ as topological interior, as illustrated.

Bracketing of the hatched set X between two subpavings. Red boxes: inner subpaving. Red and yellow: outer subpaving. The difference , outer minus inner, is a boundary approximation.
Subpavings which bracket a set with a low resolution
Subpavings which bracket the same set with a moderate resolution
Subpavings which bracket the set with a high resolution