Given a nonempty set
E \subset {\mathbb{R}}^n
E
⊂
R
n
, we provide necessary and sufficient conditions for the existence of a convex set
K \subset {\mathbb{R}}^n
K
⊂
R
n
(possibly, nonclosed and unbounded) such that
\mathrm{ext\,}K = E
e
x
t
K
=
E
. Also, we describe a family of convex sets
K \subset {\mathbb{R}}^n
K
⊂
R
n
satisfying the equality
K = \mathrm{conv\,}(\mathrm{ext\,}K)
K
=
c
o
n
v
(
e
x
t
K
)
, and, more general,
K = \mathrm{conv\,}(\mathrm{ext\,}K) + \mathrm{rec\,}K
K
=
c
o
n
v
(
e
x
t
K
)
+
r
e
c
K
, where
\mathrm{rec\,}K
r
e
c
K
denotes the recession cone of
K
K
.