Fonction d'appui

In mathematics, the support function hA of a non-empty closed convex set A in describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on . Any non-empty closed convex set A is uniquely determined by hA. Furthermore, the support function, as a function of the set A, is compatible with many natural geometric operations, like scaling, translation, rotation and Minkowski addition. Due to these properties, the support function is one of the most central basic concepts in convex geometry. The support function of a non-empty closed convex set A in is given by see Its interpretation is most intuitive when x is a unit vector: by definition, A is contained in the closed half space and there is at least one point of A in the boundary of this half space. The hyperplane H(x) is therefore called a supporting hyperplane with exterior (or outer) unit normal vector x. The word exterior is important here, as the orientation of x plays a role, the set H(x) is in general different from H(-x). Now hA is the (signed) distance of H(x) from the origin. The support function of a singleton A={a} is . The support function of the Euclidean unit ball is where is the 2-norm. If A is a line segment through the origin with endpoints -a and a then . The support function of a compact nonempty convex set is real valued and continuous, but if the set is closed and unbounded, its support function is extended real valued (it takes the value ). As any nonempty closed convex set is the intersection of its supporting half spaces, the function hA determines A uniquely. This can be used to describe certain geometric properties of convex sets analytically. For instance, a set A is point symmetric with respect to the origin if and only if hA is an even function. In general, the support function is not differentiable. However, directional derivatives exist and yield support functions of support sets.