Convex Sets
Levent Kandiller
Industrial Engineering Department
Çankaya University, Turkey
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.1/16
Convexity
Definition. A set X in Rn is said to be convex if
∀x1 , x2 ∈ X and ∀α ∈ R+ , 0 < α < 1, the point αx1 +(1−α)x2 ∈ X.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.2/16
Convexity
Definition. A set X in Rn is said to be convex if
∀x1 , x2 ∈ X and ∀α ∈ R+ , 0 < α < 1, the point αx1 +(1−α)x2 ∈ X.
x1
x2
CONVEX
c
x1
x2
NON-CONVEX
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.2/16
Convexity
Definition. A set X in Rn is said to be convex if
∀x1 , x2 ∈ X and ∀α ∈ R+ , 0 < α < 1, the point αx1 +(1−α)x2 ∈ X.
x1
x2
CONVEX
x1
x2
NON-CONVEX
Remark. Geometrically speaking, X is convex if for any points
x1 , x2 ∈ X , the line segment joining these two points is also in the set.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.2/16
Extreme point
Definition. A point x
∈ X is an extreme point of the convex set X if
and only if
6 ∃ x1 , x2 (x1 6= x2 ) ∈ X ∋ x = (1 − α)x1 + αx2 , 0 < α < 1.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.3/16
Extreme point
Definition. A point x
∈ X is an extreme point of the convex set X if
and only if
6 ∃ x1 , x2 (x1 6= x2 ) ∈ X ∋ x = (1 − α)x1 + αx2 , 0 < α < 1.
Proposition. Any extreme point is on boundary of the set.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.3/16
Extreme point
Definition. A point x
∈ X is an extreme point of the convex set X if
and only if
6 ∃ x1 , x2 (x1 6= x2 ) ∈ X ∋ x = (1 − α)x1 + αx2 , 0 < α < 1.
Proposition. Any extreme point is on boundary of the set.
Proof. Let x0 be any interior point of X . Then ∃ǫ > 0 ∋ every point in this ǫ
neighborhood of x0 is in this set. Let x1
Consider x2
= −x1 + 2x0 , |x2 − x0 | = |x1 − x0 |, then x2 is in ǫ neighborhood.
Furthermore, x0
c
6= x0 be a point in this ǫ neighborhood.
= 12 (x1 + x2 ); hence, x0 is not an extreme point.
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.3/16
Extreme point
Definition. A point x
∈ X is an extreme point of the convex set X if
and only if
6 ∃ x1 , x2 (x1 6= x2 ) ∈ X ∋ x = (1 − α)x1 + αx2 , 0 < α < 1.
Proposition. Any extreme point is on boundary of the set.
Proof. Let x0 be any interior point of X . Then ∃ǫ > 0 ∋ every point in this ǫ
neighborhood of x0 is in this set. Let x1
Consider x2
6= x0 be a point in this ǫ neighborhood.
= −x1 + 2x0 , |x2 − x0 | = |x1 − x0 |, then x2 is in ǫ neighborhood.
Furthermore, x0
= 12 (x1 + x2 ); hence, x0 is not an extreme point.
Remark. Not all boundary points of a convex set are necessarily
extreme points; they may lie between two other boundary points.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.3/16
Convexity
Proposition.
Convex sets in Rn satisfy the following relations.
If X is a convex set and β
∈ R, the set βX = {y : y = βx, x ∈ X} is
convex.
Proof.
2X
X
X+Y
X
X
Y
Y
(i)
c
(ii)
(iii)
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.4/16
Convexity
Proposition.
Convex sets in Rn satisfy the following relations.
If X is a convex set and β
∈ R, the set βX = {y : y = βx, x ∈ X} is
convex.
If X and Y are convex sets, then the set
X + Y = {z : z = x + y, x ∈ X, y ∈ Y } is convex.
Proof.
2X
X
X+Y
X
X
Y
Y
(i)
c
(ii)
(iii)
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.4/16
Convexity
Proposition.
Convex sets in Rn satisfy the following relations.
If X is a convex set and β
∈ R, the set βX = {y : y = βx, x ∈ X} is
convex.
If X and Y are convex sets, then the set
X + Y = {z : z = x + y, x ∈ X, y ∈ Y } is convex.
The intersection of any collection of convex sets is convex.
Proof.
2X
X
X+Y
X
X
Y
Y
(i)
c
(ii)
(iii)
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.4/16
Minimal Convex Set
⊂ Rn . The convex hull of S is the set which is the
intersection of all convex sets containing S .
Definition. Let S
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.5/16
Minimal Convex Set
⊂ Rn . The convex hull of S is the set which is the
intersection of all convex sets containing S .
Definition. Let S
Definition. A cone C is a set such that if x
∈ C , then αx ∈ C ,
∀α ∈ R+ . A cone which is also convex is known as convex cone.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.5/16
Minimal Convex Set
⊂ Rn . The convex hull of S is the set which is the
intersection of all convex sets containing S .
Definition. Let S
Definition. A cone C is a set such that if x
∈ C , then αx ∈ C ,
∀α ∈ R+ . A cone which is also convex is known as convex cone.
θ
θ
NON-CONVEX
c
NON-CONVEX
θ
CONVEX
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.5/16
Hyperplane
Remark. Hyperplanes dominate the entire theory of optimization;
appearing in Lagrange multipliers, duality theory, gradient calculations,
etc. The most natural definition for a hyperplane is the generalization of
a plane in R3 .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.6/16
Hyperplane
Remark. Hyperplanes dominate the entire theory of optimization;
appearing in Lagrange multipliers, duality theory, gradient calculations,
etc. The most natural definition for a hyperplane is the generalization of
a plane in R3 .
Definition. A set V in Rn is said to be linear variety, if, given any
x1 , x2 ∈ V , we have αx1 + (1 − α)x2 ∈ V, ∀α ∈ R.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.6/16
Hyperplane
Remark. Hyperplanes dominate the entire theory of optimization;
appearing in Lagrange multipliers, duality theory, gradient calculations,
etc. The most natural definition for a hyperplane is the generalization of
a plane in R3 .
Definition. A set V in Rn is said to be linear variety, if, given any
x1 , x2 ∈ V , we have αx1 + (1 − α)x2 ∈ V, ∀α ∈ R.
Remark. The only difference between a linear variety and a convex set
is that a linear variety is the entire line passing through any two points,
rather than a simple line segment.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.6/16
Hyperplane
Definition. A hyperplane in Rn is an (n − 1)-dimensional linear
variety. It can be regarded as the largest linear variety in a space other
than the entire space itself.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.7/16
Hyperplane
Definition. A hyperplane in Rn is an (n − 1)-dimensional linear
variety. It can be regarded as the largest linear variety in a space other
than the entire space itself.
∈ Rn , a 6= θ and b ∈ R. The set
H = {x ∈ Rn : aT x = b}
is a hyperplane in Rn .
Proposition. Let a
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.7/16
Hyperplane
Definition. A hyperplane in Rn is an (n − 1)-dimensional linear
variety. It can be regarded as the largest linear variety in a space other
than the entire space itself.
∈ Rn , a 6= θ and b ∈ R. The set
H = {x ∈ Rn : aT x = b}
is a hyperplane in Rn .
Proposition. Let a
Proof. Let x1 ∈ H . Translate H by −x1 , we then obtain the set
M = H − x1 = {y ∈ Rn : ∃x ∈ H ∋ y = x − x1 },
which is a linear subspace of Rn .
orthogonal vectors to a
c
M = {y ∈ Rn : aT y = 0} is also the set of all
∈ Rn , which is clearly (n − 1) dimensional.
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.7/16
Hyperplane
Proposition. Let H be an hyperplane in Rn . Then,
∃a ∈ Rn ∋ H = {x ∈ R : aT x = b}.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.8/16
a
Hyperplane
H
Proposition. Let H be an hyperplane in Rn . Then,
θ
∃a ∈ Rn ∋ H = {x ∈ R : aT x = b}.
Proof. Let x1 ∈ H , and translate by −x1 obtaining M = H − x1 . Since H is a
hyperplane, M is an (n − 1)-dimensional space.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.8/16
a
Hyperplane
H
Proposition. Let H be an hyperplane in Rn . Then,
θ
∃a ∈ Rn ∋ H = {x ∈ R : aT x = b}.
Proof. Let x1 ∈ H , and translate by −x1 obtaining M = H − x1 . Since H is a
hyperplane, M is an (n − 1)-dimensional space. Let a be any orthogonal to M , i.e.
a ∈ M ⊥ . Thus, M = {y ∈ Rn : aT y = 0}.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.8/16
a
Hyperplane
H
Proposition. Let H be an hyperplane in Rn . Then,
θ
∃a ∈ Rn ∋ H = {x ∈ R : aT x = b}.
Proof. Let x1 ∈ H , and translate by −x1 obtaining M = H − x1 . Since H is a
hyperplane, M is an (n − 1)-dimensional space. Let a be any orthogonal to M , i.e.
a ∈ M ⊥ . Thus, M = {y ∈ Rn : aT y = 0}. Let b = aT x1 we see that if
x2 ∈ H , x2 − x1 ∈ M and therefore aT x2 − aT x1 = 0 ⇒ aT x2 = b. Hence,
H ⊂ {x ∈ R : aT x = b}.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.8/16
a
Hyperplane
H
Proposition. Let H be an hyperplane in Rn . Then,
θ
∃a ∈ Rn ∋ H = {x ∈ R : aT x = b}.
Proof. Let x1 ∈ H , and translate by −x1 obtaining M = H − x1 . Since H is a
hyperplane, M is an (n − 1)-dimensional space. Let a be any orthogonal to M , i.e.
a ∈ M ⊥ . Thus, M = {y ∈ Rn : aT y = 0}. Let b = aT x1 we see that if
x2 ∈ H , x2 − x1 ∈ M and therefore aT x2 − aT x1 = 0 ⇒ aT x2 = b. Hence,
H ⊂ {x ∈ R : aT x = b}. Since H is, by definition, of (n − 1) dimension, and
{x ∈ R : aT x = b} is of dimension (n − 1) by the above proposition, these two
sets must be equal.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.8/16
Half space
∈ Rn , b ∈ R. Corresponding to the hyperplane
H = {x : aT x = b}, there are positive and negative closed half
Definition. Let a
spaces:
H+ = {x : aT x ≥ b}, H− = {x : aT x ≤ b}
and
H˙+ = {x : aT x > b}, H˙− = {x : aT x < b}.
Half spaces are convex sets and H+
c
∪ H− = R n .
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.9/16
Half space
∈ Rn , b ∈ R. Corresponding to the hyperplane
H = {x : aT x = b}, there are positive and negative closed half
Definition. Let a
spaces:
H+ = {x : aT x ≥ b}, H− = {x : aT x ≤ b}
and
H˙+ = {x : aT x > b}, H˙− = {x : aT x < b}.
Half spaces are convex sets and H+
∪ H− = R n .
Definition. A set which can be expressed as the intersection of a finite
number of closed half spaces is said to be a convex polyhedron.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.9/16
Half space
Convex polyhedra are the sets obtained as the family of
solutions to a set of linear inequalities of the form
aT1 x ≤ b1
aT2 x ≤ b2
..
.
aTm x ≤ bm
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.10/16
Half space
Convex polyhedra are the sets obtained as the family of
solutions to a set of linear inequalities of the form
aT1 x ≤ b1
aT2 x ≤ b2
..
.
aTm x ≤ bm
Since each individual entry defines a half space and the
solution family is the intersection of these half spaces.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.10/16
Half space
Convex polyhedra are the sets obtained as the family of
solutions to a set of linear inequalities of the form
aT1 x ≤ b1
aT2 x ≤ b2
..
.
aTm x ≤ bm
Since each individual entry defines a half space and the
solution family is the intersection of these half spaces.
Definition. A nonempty bounded polyhedron is called a
polytope.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.10/16
Separating Hyperplane
Theorem (Separating Hyperplane ). Let X be a convex set and y be a point exterior
to the closure of X . Then, there exists a vector a
∈ Rn ∋ aT y < inf x∈X aT x.
(Geometrically, a given point y outside X , a separating hyperplane can be passed
through the point y that does not touch X ).
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.11/16
ax=b
δ
Separating Hyperplane
y
δ
δ
a
x0 z
Theorem (Separating Hyperplane ). Let X be a convex set and y be a point exterior
to the closure of X . Then, there exists a vector a
X
∈ Rn ∋ aT y < inf x∈X aT x.
(Geometrically, a given point y outside X , a separating hyperplane can be passed
through the point y that does not touch X ).
Proof.
Let δ
= inf x∈X |x − y| > 0 Then, there is an x0 on the boundary of X such that
|x0 − y| = δ .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.11/16
ax=b
δ
Separating Hyperplane
y
δ
δ
a
x0 z
Theorem (Separating Hyperplane ). Let X be a convex set and y be a point exterior
to the closure of X . Then, there exists a vector a
X
∈ Rn ∋ aT y < inf x∈X aT x.
(Geometrically, a given point y outside X , a separating hyperplane can be passed
through the point y that does not touch X ).
Proof.
Let δ
= inf x∈X |x − y| > 0 Then, there is an x0 on the boundary of X such that
|x0 − y| = δ . Let z ∈ X . Then, ∀α, 0 ≤ α ≤ 1, x0 + α(z − x0 ) is the line segment between
x0 and z .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.11/16
ax=b
δ
Separating Hyperplane
y
δ
δ
a
x0 z
Theorem (Separating Hyperplane ). Let X be a convex set and y be a point exterior
to the closure of X . Then, there exists a vector a
X
∈ Rn ∋ aT y < inf x∈X aT x.
(Geometrically, a given point y outside X , a separating hyperplane can be passed
through the point y that does not touch X ).
Proof.
Let δ
= inf x∈X |x − y| > 0 Then, there is an x0 on the boundary of X such that
|x0 − y| = δ . Let z ∈ X . Then, ∀α, 0 ≤ α ≤ 1, x0 + α(z − x0 ) is the line segment between
x0 and z . Thus, by definition of x0 , |x0 + α(z − x0 ) − y|2 ≥ |x0 − y|2 ⇔
(x0 − y)T (x0 − y) + 2α(x0 − y)T (z − x0 ) + α2 (z − x0 )T (z − x0 ) ≥ (x0 − y)T (x0 − y)
⇔ 2α(x0 − y)T (z − x0 ) + α2 |z − x0 |2 ≥ 0
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.11/16
ax=b
δ
Separating Hyperplane
y
δ
δ
a
x0 z
Theorem (Separating Hyperplane ). Let X be a convex set and y be a point exterior
to the closure of X . Then, there exists a vector a
X
∈ Rn ∋ aT y < inf x∈X aT x.
(Geometrically, a given point y outside X , a separating hyperplane can be passed
through the point y that does not touch X ).
Proof.
Let δ
= inf x∈X |x − y| > 0 Then, there is an x0 on the boundary of X such that
|x0 − y| = δ . Let z ∈ X . Then, ∀α, 0 ≤ α ≤ 1, x0 + α(z − x0 ) is the line segment between
x0 and z . Thus, by definition of x0 , |x0 + α(z − x0 ) − y|2 ≥ |x0 − y|2 ⇔
(x0 − y)T (x0 − y) + 2α(x0 − y)T (z − x0 ) + α2 (z − x0 )T (z − x0 ) ≥ (x0 − y)T (x0 − y)
⇔ 2α(x0 − y)T (z − x0 ) + α2 |z − x0 |2 ≥ 0 Let α → 0+ , then α2 tends to 0 more rapidly than
2α. Thus, (x0 − y)T (z − x0 ) ≥ 0 ⇔ (x0 − y)T z − (x0 − y)T x0 ≥ 0
⇔ (x0 − y)T z ≥ (x0 − y)T x0 = (x0 − y)T y + (x0 + y)T (x0 − y) = (x0 − y)T y + δ 2
⇔ (x0 − y)T y < (x0 − y)T x0 ≤ (x0 − y)T z, ∀z ∈ X (Since δ > 0).
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.11/16
ax=b
δ
Separating Hyperplane
y
δ
δ
a
x0 z
Theorem (Separating Hyperplane ). Let X be a convex set and y be a point exterior
to the closure of X . Then, there exists a vector a
X
∈ Rn ∋ aT y < inf x∈X aT x.
(Geometrically, a given point y outside X , a separating hyperplane can be passed
through the point y that does not touch X ).
Proof.
Let δ
= inf x∈X |x − y| > 0 Then, there is an x0 on the boundary of X such that
|x0 − y| = δ . Let z ∈ X . Then, ∀α, 0 ≤ α ≤ 1, x0 + α(z − x0 ) is the line segment between
x0 and z . Thus, by definition of x0 , |x0 + α(z − x0 ) − y|2 ≥ |x0 − y|2 ⇔
(x0 − y)T (x0 − y) + 2α(x0 − y)T (z − x0 ) + α2 (z − x0 )T (z − x0 ) ≥ (x0 − y)T (x0 − y)
⇔ 2α(x0 − y)T (z − x0 ) + α2 |z − x0 |2 ≥ 0 Let α → 0+ , then α2 tends to 0 more rapidly than
2α. Thus, (x0 − y)T (z − x0 ) ≥ 0 ⇔ (x0 − y)T z − (x0 − y)T x0 ≥ 0
⇔ (x0 − y)T z ≥ (x0 − y)T x0 = (x0 − y)T y + (x0 + y)T (x0 − y) = (x0 − y)T y + δ 2
⇔ (x0 − y)T y < (x0 − y)T x0 ≤ (x0 − y)T z, ∀z ∈ X (Since δ > 0).
Let a
= (x0 − y), then aT y < aT x0 = inf z∈X aT z .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.11/16
Supporting Hyperplane
Theorem (Supporting Hyperplane ). Let X be a convex set, and let y
be a boundary point of X . Then, there is a hyperplane containing y and
containing X in one of its closed half spaces.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.12/16
Supporting Hyperplane
Theorem (Supporting Hyperplane ). Let X be a convex set, and let y
be a boundary point of X . Then, there is a hyperplane containing y and
containing X in one of its closed half spaces.
Proof. Let{yk } be sequence of vectors, exterior to the closure of X , converging to y .
Let {ak } be a sequence of corresponding vectors constructed according to the
= 1, such that aTk yk < inf x∈X . Since
{ak } is a boundary sequence, it converges to a. For this vector, we have
aT y = lim aTk yk ≤ ax.
previous theorem, normalized so that |ak |
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.12/16
Supporting Hyperplane
Theorem (Supporting Hyperplane ). Let X be a convex set, and let y
be a boundary point of X . Then, there is a hyperplane containing y and
containing X in one of its closed half spaces.
Proof. Let{yk } be sequence of vectors, exterior to the closure of X , converging to y .
Let {ak } be a sequence of corresponding vectors constructed according to the
= 1, such that aTk yk < inf x∈X . Since
{ak } is a boundary sequence, it converges to a. For this vector, we have
aT y = lim aTk yk ≤ ax.
previous theorem, normalized so that |ak |
Definition. A hyperplane containing a convex set X in one of its closed
half spaces and containing a boundary point of X is said to be
supporting hyperplane of X.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.12/16
Extreme points
Remark. We have already defined extreme points. For example, the
extreme points of a square are its corners in R2 whereas the extreme
points of a circular disk are all (infinitely many!) the points on the
boundary circle. Note that, a linear variety consisting of more than one
point has no extreme points.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.13/16
Extreme points
Lemma. Let X be a convex set, H be a supporting
hyperplane of X and T = X ∩ H . Every extreme point of
T is an extreme point of X .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.13/16
Extreme points
Lemma. Let X be a convex set, H be a supporting
hyperplane of X and T = X ∩ H . Every extreme point of
T is an extreme point of X .
Proof. Suppose x0 ∈ T is not an extreme point of X . Then,
x0 = αx1 + (1 − α)x2 for some x1 , x2 ∈ X, 0 < α < 1.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.13/16
Extreme points
Lemma. Let X be a convex set, H be a supporting
hyperplane of X and T = X ∩ H . Every extreme point of
T is an extreme point of X .
Proof. Suppose x0 ∈ T is not an extreme point of X . Then,
x0 = αx1 + (1 − α)x2 for some x1 , x2 ∈ X, 0 < α < 1.
Let H
= {x : aT x = c} with X contained in its closed positive half space. Then,
aT x1 ≥ c, aT x2 ≥ c.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.13/16
Extreme points
Lemma. Let X be a convex set, H be a supporting
hyperplane of X and T = X ∩ H . Every extreme point of
T is an extreme point of X .
Proof. Suppose x0 ∈ T is not an extreme point of X . Then,
x0 = αx1 + (1 − α)x2 for some x1 , x2 ∈ X, 0 < α < 1.
Let H
= {x : aT x = c} with X contained in its closed positive half space. Then,
aT x1 ≥ c, aT x2 ≥ c. However, since x0 ∈ H ,
c = aT x0 = αaT x1 + (1 − α)aT x2 .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.13/16
Extreme points
Lemma. Let X be a convex set, H be a supporting
hyperplane of X and T = X ∩ H . Every extreme point of
T is an extreme point of X .
Proof. Suppose x0 ∈ T is not an extreme point of X . Then,
x0 = αx1 + (1 − α)x2 for some x1 , x2 ∈ X, 0 < α < 1.
Let H
= {x : aT x = c} with X contained in its closed positive half space. Then,
aT x1 ≥ c, aT x2 ≥ c. However, since x0 ∈ H ,
c = aT x0 = αaT x1 + (1 − α)aT x2 .
Thus, x1 , x2
c
∈ H.
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.13/16
Extreme points
Lemma. Let X be a convex set, H be a supporting
hyperplane of X and T = X ∩ H . Every extreme point of
T is an extreme point of X .
Proof. Suppose x0 ∈ T is not an extreme point of X . Then,
x0 = αx1 + (1 − α)x2 for some x1 , x2 ∈ X, 0 < α < 1.
Let H
= {x : aT x = c} with X contained in its closed positive half space. Then,
aT x1 ≥ c, aT x2 ≥ c. However, since x0 ∈ H ,
c = aT x0 = αaT x1 + (1 − α)aT x2 .
Thus, x1 , x2
c
∈ H . Hence, x1 , x2 ∈ T and x0 is not an extreme point of T .
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.13/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
For n
= 1, the statement is true for a line segment:
[a, b] = {x ∈ R : x = αa + (1 − α)b, 0 ≤ α ≤ 1}.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Suppose that the theorem is true for (n − 1). Let X be a closed bounded convex set in Rn , and let K be
the convex hull of the extreme points of X . We will show that X
c
= K.
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x. Let x0 = inf x∈X (aT x). x0 is finite and
∃x0 ∈ X ∋ aT x0 = b0
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x. Let x0 = inf x∈X (aT x). x0 is finite and
∃x0 ∈ X ∋ aT x0 = b0
(because by Weierstrass’ Theorem: The continuous function aT x achieve its
minimum over any closed bounded set).
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x. Let x0 = inf x∈X (aT x). x0 is finite and
∃x0 ∈ X ∋ aT x0 = b0
Hence, the hyperplane H
= {x : aT x = b0 } is a supporting hyperplane
to X .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x. Let x0 = inf x∈X (aT x). x0 is finite and
∃x0 ∈ X ∋ aT x0 = b0
to X .
Since b0
c
Hence, the hyperplane H
= {x : aT x = b0 } is a supporting hyperplane
≤ aT y ≤ inf x∈K aT x, H is disjoint from K .
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x. Let x0 = inf x∈X (aT x). x0 is finite and
∃x0 ∈ X ∋ aT x0 = b0
to X .
Since b0
Hence, the hyperplane H
= {x : aT x = b0 } is a supporting hyperplane
≤ aT y ≤ inf x∈K aT x, H is disjoint from K . Let T = H ∩ X . Then, T is a
bounded closed convex set of H , which can be regarded as a space in Rn−1 .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x. Let x0 = inf x∈X (aT x). x0 is finite and
∃x0 ∈ X ∋ aT x0 = b0
to X .
Since b0
Hence, the hyperplane H
= {x : aT x = b0 } is a supporting hyperplane
≤ aT y ≤ inf x∈K aT x, H is disjoint from K . Let T = H ∩ X . Then, T is a
bounded closed convex set of H , which can be regarded as a space in Rn−1 .
c
T 6= ∅, since x0 ∈ T .
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x. Let x0 = inf x∈X (aT x). x0 is finite and
∃x0 ∈ X ∋ aT x0 = b0
to X .
Since b0
Hence, the hyperplane H
= {x : aT x = b0 } is a supporting hyperplane
≤ aT y ≤ inf x∈K aT x, H is disjoint from K . Let T = H ∩ X . Then, T is a
bounded closed convex set of H , which can be regarded as a space in Rn−1 .
T 6= ∅, since x0 ∈ T .
Hence, by induction hypothesis, T contains extreme points; and by the previous Lemma, these are the
extreme points of X .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x. Let x0 = inf x∈X (aT x). x0 is finite and
∃x0 ∈ X ∋ aT x0 = b0
to X .
Since b0
Hence, the hyperplane H
= {x : aT x = b0 } is a supporting hyperplane
≤ aT y ≤ inf x∈K aT x, H is disjoint from K . Let T = H ∩ X . Then, T is a
bounded closed convex set of H , which can be regarded as a space in Rn−1 .
T 6= ∅, since x0 ∈ T .
Hence, by induction hypothesis, T contains extreme points; and by the previous Lemma, these are the
extreme points of X .
c
Thus, we have found extreme points of X not in K , Contradiction.
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Extreme points
Theorem. A closed bounded convex set in Rn is equal to the closed
convex hull of its extreme points.
Proof. This proof is by induction on n.
Assume that ∃y
and K ; ∃a
∈ X ∋ y 6∈ K . Then, by the Separating Theorem, there is a hyperplane separating y
6= 0 ∋ aT y < inf x∈K aT x. Let x0 = inf x∈X (aT x). x0 is finite and
∃x0 ∈ X ∋ aT x0 = b0
to X .
Since b0
Hence, the hyperplane H
= {x : aT x = b0 } is a supporting hyperplane
≤ aT y ≤ inf x∈K aT x, H is disjoint from K . Let T = H ∩ X . Then, T is a
bounded closed convex set of H , which can be regarded as a space in Rn−1 .
T 6= ∅, since x0 ∈ T .
Hence, by induction hypothesis, T contains extreme points; and by the previous Lemma, these are the
extreme points of X .
Thus, we have found extreme points of X not in K , Contradiction.
Therefore,
X ⊆ K , and hence X = K (since K ⊆ X , i.e. K is closed and bounded).
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.14/16
Convex Polytope
Remark. Let us investigate the implications of this theorem for convex
polytopes. A convex polytope is a bounded polyhedron. Being the
intersection of closed halfspaces, a convex polytope is closed. Thus,
any convex polyhedron is the closed convex hull of its extreme points. It
can be shown that any polytope has at most a finite number of extreme
points, and hence a convex polytope is equal to the convex hull of a
finite number of points. The converse can also be established, yielding
the following two equivalent characterizations.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.15/16
Convex Polytope
Remark. Let us investigate the implications of this theorem for convex
polytopes. A convex polytope is a bounded polyhedron. Being the
intersection of closed halfspaces, a convex polytope is closed. Thus,
any convex polyhedron is the closed convex hull of its extreme points. It
can be shown that any polytope has at most a finite number of extreme
points, and hence a convex polytope is equal to the convex hull of a
finite number of points. The converse can also be established, yielding
the following two equivalent characterizations.
Theorem. A convex polytope can be described either as a
bounded intersection of a finite number of closed half
spaces, or as the convex hull of a finite number of points.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.15/16
Collaborative Work: Polytopes
Characterize zero dimensional polytopes.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Characterize zero dimensional polytopes.
A zero dimensional polytope is a point.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Characterize zero dimensional polytopes.
A zero dimensional polytope is a point.
Characterize one dimensional polytopes.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Characterize zero dimensional polytopes.
A zero dimensional polytope is a point.
Characterize one dimensional polytopes.
One dimensional polytopes are line segments.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Characterize zero dimensional polytopes.
A zero dimensional polytope is a point.
Characterize one dimensional polytopes.
One dimensional polytopes are line segments.
Characterize two dimensional polytopes.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Characterize zero dimensional polytopes.
A zero dimensional polytope is a point.
Characterize one dimensional polytopes.
One dimensional polytopes are line segments.
Characterize two dimensional polytopes.
Two dimensional polytopes are n-gons: triangle (3), rectangle (4),
trapezoid (4), pentagon (5), . . .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
d-simplex is the convex hull of any d + 1 independent points in Rn
(n ≥ d). Standard d − simplex with d + 1 vertices in Rd+1 is
Pd+1
d+1
∆d = {x ∈ R
: i=1 xi = 1; xi ≥ 0, i = 1, . . . , d + 1}.
Characterize ∆2 in R3 .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
3
d-simplex is the convex hull of any d + 1 independent points in Rn
(n ≥ d). Standard d − simplex with d + 1 vertices in Rd+1 is
Pd+1
d+1
∆d = {x ∈ R
: i=1 xi = 1; xi ≥ 0, i = 1, . . . , d + 1}.
Characterize ∆2 in R3 .
∆2 =conv(e1 , e2 , e3 ).
(0,0,1)
X
(1,1,1)
c
(1,0,0)
X1
2
X
(0,1,0)
.
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Characterize cubes and octahedrons with the help of three
dimensional cube C3 , and octahedron C3∆ .
CUBE
c
OCTAHEDRON
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Characterize cubes and octahedrons with the help of three
dimensional cube C3 , and octahedron C3∆ .
CUBE
OCTAHEDRON
CUBE:
C3 =conv((0, 0, 0)T , (α, 0, 0)T , (0, α, 0)T , (0, 0, α)T , (α, α, 0)T , (α, 0, α)T , (0, α, α)T , (α, α, α)T )
Cn = {x ∈ Rn : 0 ≤ xi ≤ α, i = 1, . . . , n; α ∈ R+ } .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Characterize cubes and octahedrons with the help of three
dimensional cube C3 , and octahedron C3∆ .
CUBE
OCTAHEDRON
OCTAHEDRON:
C3∆ =conv((α, 0, 0)T , (0, α, 0)T , (0, 0, α)T , (−α, 0, 0)T , (0, −α, 0)T , (0, 0, −α)T )
Pn
∆
n
Cn = {x ∈ R :
i=1 |xi | ≤ α, α ∈ R+ } .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
describing inequalities.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
c
2
X
F0
(1,0,0)
(1,1,0)
X1
F2
F3
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
2
X
F0
(1,0,0)
(1,1,0)
X1
F2
F3
Let ai be the normal to face Fi , i = 0, 1, 2, 3, 4. Let ai x ≤ bi be the respective defining inequalities.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
2
X
F0
(1,0,0)
(1,1,0)
F3
X1
F2
We know F0 is the x1 –x2 plane. Then, F0 = x ∈ R3 : x3 = 0 .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
2
X
F0
(1,0,0)
(1,1,0)
X1
F2
F3
We know that a2 and a4 are perpendicular to x2 –axis as a1 and a3 are perpendicular to x1 –axis. Thus,
a1 = (0, ∗, ∗)T , a2 = (∗, 0, ∗)T , a3 = (0, ∗, ∗)T , a4 = (∗, 0, ∗)T .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
2
X
F0
(1,0,0)
(1,1,0)
F3
X1
F2
Since F1 contains (1/2, 1/2, 1), (1, 0, 0), (0, 0, 0), F1 = x ∈ R3 : 0x1 − 2x2 + 1x3 = 0 .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
2
X
F0
(1,0,0)
(1,1,0)
F3
X1
F2
Since F2 contains (1/2, 1/2, 1), (1, 0, 0), (1, 1, 0), F2 = x ∈ R3 : 2x1 + 0x2 + 1x3 = 2 .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
2
X
F0
(1,0,0)
(1,1,0)
F3
X1
F2
Since F3 contains (1/2, 1/2, 1), (1, 1, 0), (0, 1, 0), F3 = x ∈ R3 : 0x1 + 2x2 + 1x3 = 2 .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
2
X
F0
(1,0,0)
(1,1,0)
F3
X1
F2
And finally, (1/2, 1/2, 1), (0, 1, 0), (0, 0, 0) are in F4 = x ∈ R3 : −2x1 + 0x2 + 1x3 = 0 .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
2
X
F0
(1,0,0)
(1,1,0)
X1
F2
F3
Therefore,
P3 = {x ∈ R3 : x3 ≥ 0, −2x2 + x3 ≤ 0, 2x1 + x3 ≤ 2, 2x2 + x3 ≤ 2, −2x1 + x3 ≤ 0}.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16
Collaborative Work: Polytopes
Let Pn+1 =conv(Cn , x0 ) be a (n+1)–dimensional pyramid, where
x0 6∈ Cn . Draw P3 = conv(C2 : α = 1, (1/2, 1/2, 1)T ) and write down all
3
describing inequalities.
X
(1/2,1/2,1)
F4
F1
(0,0,0)
(0,1,0)
2
X
F0
(1,0,0)
(1,1,0)
X1
F2
F3
Pn+1 is not a union of a cone at x0 and a polytope.
Pn+1 is a direct sum of a cone at x0 and Cn .
Pn+1 is an intersection of a cone at x0 and Cn+1 provided that x0 ∈ Cn+1 \ Cn .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Convex Sets – p.16/16