Conical combination
Encyclopedia
Given a finite number of vectors 
in a real vector space, a conical combination or a conical sum of these vectors is a vector of the form
where the real numbers
satisfy 
The name derives from the fact that a conical sum of vectors defines a cone
(possibly in a lower dimensional subspace).
By definition, the zero point (origin
) belongs to all conical hulls.
The conical hull of a set S is a convex set
. In fact, it is the intersection of all convex cone
s containing S plus the origin. If S is a compact set (in particular, when it is a finite set of points), then the condition "plus the origin" is unnecessary.
If we discard the origin, we can divide all coefficients by their sum to see that a conical combination is a convex combination
scaled by a positive factor.
Therefore, the "conical combination" and "conical hull" are more accurately to be called the "convex conical combination" and "convex conical hull" respectively. Moreover, the above remark about dividing the coefficients while discarding the origin implies that the conical combinations and hulls may be considered as convex combinations and convex hull
s in the projective space
.
While the convex hull of a compact set is a compact set as well, this is not so for the conical hull: first of all, the latter one is unbounded. Moreover, it is even not necessarily a closed set
: a counterexample is a sphere
passing through the origin, with the conical hull being an open half-space
plus the origin. However if S is a nonempty compact set which does not contain the origin, the conical hull is a closed set.
in a real vector space, a conical combination or a conical sum of these vectors is a vector of the form
where the real numbers
satisfy The name derives from the fact that a conical sum of vectors defines a cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
(possibly in a lower dimensional subspace).
Conical hull
The set of all conical combinations for a given set S is called the conical hull of S and denoted cone (S), or coni (S), that is,By definition, the zero point (origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
) belongs to all conical hulls.
The conical hull of a set S is a convex set
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
. In fact, it is the intersection of all convex cone
Convex cone
In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...
s containing S plus the origin. If S is a compact set (in particular, when it is a finite set of points), then the condition "plus the origin" is unnecessary.
If we discard the origin, we can divide all coefficients by their sum to see that a conical combination is a convex combination
Convex combination
In convex geometry, a convex combination is a linear combination of points where all coefficients are non-negative and sum up to 1....
scaled by a positive factor.
Therefore, the "conical combination" and "conical hull" are more accurately to be called the "convex conical combination" and "convex conical hull" respectively. Moreover, the above remark about dividing the coefficients while discarding the origin implies that the conical combinations and hulls may be considered as convex combinations and convex hull
Convex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
s in the projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
.
While the convex hull of a compact set is a compact set as well, this is not so for the conical hull: first of all, the latter one is unbounded. Moreover, it is even not necessarily a closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
: a counterexample is a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
passing through the origin, with the conical hull being an open half-space
Half-space
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional euclidean space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space...
plus the origin. However if S is a nonempty compact set which does not contain the origin, the conical hull is a closed set.
Related combinations
- Affine combination
- Convex combinationConvex combinationIn convex geometry, a convex combination is a linear combination of points where all coefficients are non-negative and sum up to 1....
- Linear combinationLinear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...