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Given a finite number of vectors in a
pp. 101, 102
/ref>''Mathematical Programming'', by Melvyn W. Jeter (1986)
p. 68
/ref> of these vectors is a vector of the form : where are
Therefore, "conical combinations" and "conical hulls" are in fact "convex conical combinations" and "convex conical hulls" 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 hulls in the projective space.
While the convex hull of a compact set is also a compact set, this is not so for the conical hull; first of all, the latter one is unbounded. Moreover, it is not even necessarily a closed set: a counterexample is a
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vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102
/ref>''Mathematical Programming'', by Melvyn W. Jeter (1986)
p. 68
/ref> of these vectors is a vector of the form : where are
non-negative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
real numbers.
The name derives from the fact that a conical sum of vectors defines a cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
(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 taking ''k'' = 0, it follows the zero vector (origin
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) belongs to all conical hulls (since the summation becomes an empty sum
In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero.
The natural way to extend non-empty sums is to let the empty sum be the additive identity.
Let a_1, a_2, a_3, ... be a sequence of numbers, and let
...
).
The conical hull of a set ''S'' is a convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
. In fact, it is the intersection of all convex cones containing ''S'' plus the origin. If ''S'' is a compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
(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
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other w ...
scaled by a positive factor.
Therefore, "conical combinations" and "conical hulls" are in fact "convex conical combinations" and "convex conical hulls" 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 hulls in the projective space.
While the convex hull of a compact set is also a compact set, this is not so for the conical hull; first of all, the latter one is unbounded. Moreover, it is not even necessarily a closed set: a counterexample is a sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
passing through the origin, with the conical hull being an open half-space plus the origin. However, if ''S'' is a non-empty convex compact set which does not contain the origin, then the convex conical hull of ''S'' is a closed set.
See also
Related combinations
*Affine combination In mathematics, an affine combination of is a linear combination
: \sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_,
such that
:\sum_^ =1.
Here, can be elements ( vectors) of a vector space over a field , and the coefficients \alpha_ ...
*Convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other w ...
* Linear combination
References
{{reflist Convex geometry Mathematical analysis