Convexpolygoneille

Convexpolygoneille is a hypothetical concept in planar geometry describing a class of convex polygons equipped with a hierarchical boundary refinement structure, abbreviated as an ille-structure. The base object is a convex polygon P in the plane, with a labeling L that assigns to each boundary vertex a nonnegative integer level. Level-0 vertices are the original corners of P; higher levels arise by subdividing selected boundary edges. Each subdivision inserts a new vertex along the edge and assigns it a level equal to the parent edge’s level plus one, while preserving convexity. The collection of vertices with a given level k is used to form a convex hull P_k, giving a nested sequence of polygons that approximate P from inside as k grows.

Properties of this framework include locality of refinement, compatibility with affine transformations, and preservation of convexity at every stage. The sequence P_0, P_1, …, P_N is monotone with P_N equal to P when all boundary vertices have been assigned a level, and each P_k is contained in P_{k+1}. The hierarchical boundary representation enables multi-resolution processing, since low-level hulls capture coarse boundary structure and higher levels add detail along the edges.

Relation to convex polygons: any convex polygon can be viewed as a convexpolygoneille by choosing an appropriate level assignment, with P_N coinciding with the original polygon when all vertices are included. Conversely, a convexpolygoneille emphasizes the refinement history of the boundary, rather than the final hull alone.

Applications and usage: the concept is intended for theoretical discussions of multi-resolution representations, adaptive mesh generation, and hierarchical rendering in computational geometry. See also: convex polygon, polygon refinement, hierarchical mesh.