Tangentset

Tangentset is a concept used in geometry and geometric measure theory to describe the local, asymptotic tangential structure of a set near a point. In practice, it is often referred to as a tangent cone or a tangent set, and it can be viewed as the limit of rescaled copies of the set around the point of interest.

Formally, let E be a subset of Euclidean space R^n and x0 a point in the closure of E. A tangent set (or tangent cone) T to E at x0 is any limit, under a shrinking scale r_k → 0+, of the sets (E − x0)/r_k, where the limit is taken in a specified sense of set convergence. Equivalently, T consists of all vectors v for which there exist sequences x_k ∈ E and r_k → 0+ with x_k → x0 and (x_k − x0)/r_k → v. The tangent set is a cone with its vertex at the origin and captures the first-order, directional structure of E near x0.

Examples illustrate the concept: for a smooth curve in the plane, the tangent set at a point is the tangent line to the curve; for a smooth surface, it is the tangent plane. More singular sets, such as fractals, can yield more complex tangent sets that reveal local geometric features not visible at a coarse scale.

Related ideas include tangent space (for smooth manifolds), blow-ups, and lower/upper tangent sets. The notion is a key tool in regularity theory, geometric measure theory, and the study of singularities, where understanding all possible tangent sets at a point helps describe local behavior. Note that the term tangentset is not standard; in literature, tangent set or tangent cone is typically used.