Given a finite number of vectors

x_1, x_2, \dots, x_n

in a real

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

, 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 In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...

real numbers. The name derives from the fact that the set of all conical sum of vectors defines a

cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

(possibly in a lower-dimensional subspace).

Conical hull

The

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 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 set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

. In fact, it is the intersection of all

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

s 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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., 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 space, vector algebra, a convex combination is a linear combination of point (geometry), points (which can be vector (geometric), vectors, scalar (mathematics), scalars, or more generally points in an affine sp ...

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, 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 (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...

: a counterexample is a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

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