The cone hull is a special hull operator that assigns a cone to each subset of a vector space, more precisely the smallest cone that contains the set.
definition
Let a - vector space and an arbitrary subset of . Then is called
R.
{\ displaystyle \ mathbb {R}}
V
{\ displaystyle V}
X
{\ displaystyle X}
V
{\ displaystyle V}
cone
(
X
)
: =
⋂
X
⊆
K
K
cone
K
{\ displaystyle \ operatorname {cone} (X): = \ bigcap _ {X \ subseteq {\ mathcal {K}} \ atop {\ mathcal {K}} {\ text {cone}}} {\ mathcal {K} }}
the cone shell of . It is the smallest cone that contains.
X
{\ displaystyle X}
X
{\ displaystyle X}
The definition is equivalent to this
cone
(
X
)
: =
{
λ
x
|
x
∈
X
,
λ
≥
0
}
{\ displaystyle \ operatorname {cone} (X): = \ {\ lambda x \, | \, x \ in X, \, \ lambda \ geq 0 \}}
.
Remarks
More generally, the cone hull can be defined for any vector spaces as long as it is an ordered body .
K
{\ displaystyle \ mathbb {K}}
K
{\ displaystyle \ mathbb {K}}
The notation is not used uniformly in the literature, sometimes it also denotes the smallest convex cone that contains and is then referred to as a conical envelope or positive envelope .
cone
(
X
)
{\ displaystyle \ operatorname {cone} (X)}
X
{\ displaystyle X}
properties
cone
{\ displaystyle \ operatorname {cone}}
is an envelope operator , so it holds for
X
,
Y
⊂
V
{\ displaystyle X, Y \ subset V}
X
⊂
cone
(
X
)
{\ displaystyle X \ subset \ operatorname {cone} (X)}
,
X
⊂
Y
⟹
cone
(
X
)
⊂
cone
(
Y
)
{\ displaystyle X \ subset Y \ implies \ operatorname {cone} (X) \ subset \ operatorname {cone} (Y)}
,
cone
(
cone
(
X
)
)
=
cone
(
X
)
{\ displaystyle \ operatorname {cone} (\ operatorname {cone} (X)) = \ operatorname {cone} (X)}
.
If the convex hull of and is the conical hull , then
conv
(
X
)
{\ displaystyle \ operatorname {conv} (X)}
X
{\ displaystyle X}
pos
(
X
)
{\ displaystyle \ operatorname {pos} (X)}
cone
(
conv
(
X
)
)
=
conv
(
cone
(
X
)
)
=
pos
(
X
)
{\ displaystyle \ operatorname {cone} (\ operatorname {conv} (X)) = \ operatorname {conv} (\ operatorname {cone} (X)) = \ operatorname {pos} (X)}
.
Examples
The two vectors are given
v
1
=
(
0
1
)
,
v
2
=
(
1
0
)
{\ displaystyle v_ {1} = {\ begin {pmatrix} 0 \\ 1 \ end {pmatrix}}, \, v_ {2} = {\ begin {pmatrix} 1 \\ 0 \ end {pmatrix}}}
.
Then
cone
(
v
1
,
v
2
)
=
λ
1
(
0
1
)
∪
λ
2
(
1
0
)
,
λ
2
,
λ
2
≥
0
{\ displaystyle \ operatorname {cone} (v_ {1}, v_ {2}) = \ lambda _ {1} {\ begin {pmatrix} 0 \\ 1 \ end {pmatrix}} \ cup \ lambda _ {2} {\ begin {pmatrix} 1 \\ 0 \ end {pmatrix}}, \, \ lambda _ {2}, \ lambda _ {2} \ geq 0}
If one considers the vector space of the matrices as well as the set of all rotation matrices
2
×
2
{\ displaystyle 2 \ times 2}
X
{\ displaystyle X}
R.
α
=
(
cos
α
-
sin
α
sin
α
cos
α
)
{\ displaystyle R _ {\ alpha} = {\ begin {pmatrix} \ cos \ alpha & - \ sin \ alpha \\\ sin \ alpha & \ cos \ alpha \ end {pmatrix}}}
,
so is the cone of the matrices that describe
torsional elongations
cone
(
X
)
{\ displaystyle \ operatorname {cone} (X)}
cone
(
X
)
=
(
λ
cos
α
-
λ
sin
α
λ
sin
α
λ
cos
α
)
,
λ
≥
0
{\ displaystyle \ operatorname {cone} (X) = {\ begin {pmatrix} \ lambda \ cos \ alpha & - \ lambda \ sin \ alpha \\\ lambda \ sin \ alpha & \ lambda \ cos \ alpha \ end { pmatrix}}, \, \ lambda \ geq 0}
.
literature
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">