innerdiametern

innerdiametern is a geometric quantity defined for subsets of Euclidean space. It measures the largest distance between two points of the set that can be connected by a straight line segment lying entirely within the set. In other words, it captures how far apart two points can be while still allowing a direct line-of-sight between them inside the region.

Formally, for a subset A of R^n, the innerdiametern D_in(A) is the supremum of the distances ||x − y|| over all x, y ∈ A such that the line segment [x, y] is contained in A. If no two distinct points of A can be joined by such a segment, D_in(A) can be defined as 0. This convention ensures the measure remains well-defined for a wide range of shapes.

Relation to convexity is a central point: if A is convex, every pair of points in A can be joined by a segment within A, so D_in(A) equals the ordinary diameter diam(A). For non-convex sets, D_in(A) may be strictly smaller than the diameter, reflecting the presence of indentations, holes, or other obstructions that prevent long chords from lying entirely inside the set.

Examples help illustrate the concept. A disk of radius r has innerdiametern equal to 2r, the same as its diameter, because any two points can be connected by a chord inside the disk. A crescent-shaped region typically has a smaller inner diameter than its outer span, since some long straight lines through the region exit the set.

In computation and applications, the innerdiametern is relevant in shape analysis, computer graphics, and motion planning, where it reflects the extent of straight, obstacle-free connections within a region. The concept generalizes to other metric spaces and, in spaces with a notion of geodesics, to geodesic inner diameters as well.