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.
