•
NP is a
complete fan, meaning the union of its cones is the whole space, (
Rn)*. • If
F is a face of
P of dimension
d, then its normal cone
CF has dimension
n –
d. The normal cones to vertices of
P are full dimensional. If
P has full dimension, the normal cones to the facets of
P are the rays of
NP and the normal cone to
P itself is
CP = {0}, the zero cone. • The
affine span of face
F of
P is
orthogonal to the linear span of its normal cone,
CF. • The correspondence between faces of
P and cones of
NP reverses inclusion, meaning that for faces
F and
G of
P, ::F \subseteq G \quad \Leftrightarrow \quad C_F \supseteq C_G. • Since
NP is a fan, the
intersection of any two of its cones is also a cone in
NP. For faces
F and
G of
P, ::C_F \cap C_G = C_H :where
H is the smallest face of
P that contains both
F and
G. ==Applications==