Tangent Cones and Intersection Theory

Abstract

Let E be an arbitrary set in ℝ^N. A vector ν ∈ ℝ^N is called tangent to E at a point a ∈ Ē if there exist a sequence of points a^j ∈ E and numbers t _j > 0 such that a^j → a and t _j (a^j − a) → ν as j→ ∞. The set of all such tangent vectors is denoted by C(E, a) and is called the tangent cone to E at a. This really is a cone with vertex 0 since if ν ∈ C(E, a), then tν also belongs to C(E, a) for all t ≥ 0. Geometrically the cone C(E, a) is the set of limit positions of secants of E passing through a; it is the set of limit points of the family of sets t(E − a) = {t(x − a):x ∈ E} as t→∞. If a ∉ Ē then, by definition, the set C (E, a) is empty.