Cone shell

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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

the cone shell of . It is the smallest cone that contains.

The definition is equivalent to this

.

Remarks

  • More generally, the cone hull can be defined for any vector spaces as long as it is an ordered body .
  • 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 .

properties

  • is an envelope operator , so it holds for
  • ,
  • ,
  • .
  • If the convex hull of and is the conical hull , then
.

Examples

The two vectors are given

.

Then

If one considers the vector space of the matrices as well as the set of all rotation matrices

,

so is the cone of the matrices that describe torsional elongations

.

literature