The normal cone to a closed convex set 
at a point 
is
 |
In the illustration, the point 
is a vertex of the convex
polygon 
,
and the red boundary rays of 
are generated by outward normal
vectors 
and 
to the two incident sides.
For nonzero 
and 
, the defining inequality says that
the angle between them is a right
angle or an obtuse angle. The normal cone is
a convex cone. By convention, 
is the empty set when 
.
At a point in the interior of a full-dimensional 
,
the normal cone is 
.
At a smooth boundary point of a full-dimensional
, it is the ray
generated by the outward normal vector. At a vertex of a convex polyhedron,
it is generated by the outward normal vectors of
the facets incident to the vertex.
Normal cones are fundamental in the optimality conditions of convex
optimization theory.
