Given a finite number of vectors

x_1, x_2, \dots, x_n

in a real vector space, 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 :

\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n

where

\alpha_i

are non-negative real numbers. The name derives from the fact that a conical sum of vectors defines a cone (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, :

\operatorname (S)=\left\.

By taking ''k'' = 0, it follows the zero vector ( origin) belongs to all conical hulls (since the summation becomes an empty sum). The conical hull of a set ''S'' is a convex set. In fact, it is the intersection of all

convex cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . ...

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. Circle-conic-hull

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 hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

s in the

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

. 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 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. In a complete metric space, a cl ...

: 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 non-empty convex compact set which does not contain the origin, then the convex conical hull of ''S'' is a closed set.

References

{{reflist Convex geometry Mathematical analysis

Conical hull

See also

Related combinations

References