The Gauss Map of a Nonsmooth Convex Cone and the Antipodal Mate Property

We discuss some aspects concerning the angular structure of a closed convex cone

in a Euclidean vector space

E E

. The cone under consideration is assumed to be pointed and solid, but not necessarily smooth. Its Gauss map

G_K G K

is therefore to be understood in a multivalued sense. By definition,

G_K G K

assigns to a boundary point

u u

of

K K

the set

G_K(u):= N_K(u)\cap S_E G K ( u ) : = N K ( u ) ∩ S E

, where

S_E S E

is the unit sphere of

E E

and

N_K N K

is the normal cone map of

K K

in the sense of convex analysis. By a positive homogeneity argument, there is no loss of generality in assuming that

u u

has unit length. Among other issues, we elaborate on the connection between

G_K(u) G K ( u )

and the set

M_K(u) M K ( u )

of antipodal mates of

u u

. That

v v

is an antipodal mate of

u u

means that

\{u,v\} { u , v }

is a pair of unit vectors in the boundary of

K K

achieving the maximum angle of the cone.