Convex cone

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real <a href="/facts/Vector_space/M1pxMLx2">vector space</a> that is <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closed</a> under positive scalar multiplication; that is, 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a cone if 
 
 
 
 x
 ∈
 C
 
 
 {\displaystyle x\in C}
 
 implies 
 
 
 
 s
 x
 ∈
 C
 
 
 {\displaystyle sx\in C}
 
 for every positive scalar 
 
 
 
 s
 
 
 {\displaystyle s}
 
. This is a broad generalization of the standard <a href="/facts/Cone/2c9GIzYD">cone</a> in <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>.
A convex cone is a cone that is also closed under addition, or, equivalently, a subset of a vector space that is closed under <a href="/facts/Linear_combination/CVjL6kuN">linear combinations</a> with positive coefficients. It follows that convex cones are <a href="/facts/Convex_set/vdAuJRJl">convex sets</a>.
The definition of a convex cone makes sense in a vector space over any <a href="/facts/Ordered_field/an0prSgG">ordered field</a>, although the field of <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a> is used most often.

Convex cone open-in-new

Convex cone