Absolutely convex set

Absolutely convex set

A set "C" in a real or complex vector space is said to be absolutely convex if it is convex and balanced.

Properties

A set C is absolutely convex if and only if for any points x_1, , x_2 in C and any numbers lambda_1, , lambda_2 satisfying |lambda_1| + |lambda_2| leq 1 the sum lambda_1 x_1 + lambda_2 x_2belongs to C.

Since the intersection of any collection of absolutely convex sets is absolutely convex thenfor any subset "A" of a vector space one can define its absolutely convex hullto be the intersection of all absolutely convex sets containing "A".

Absolutely convex hull

The absolutely convex hull of the set "A" assumes the following representation

mbox{absconv} A = left{sum_{i=1}^nlambda_i x_i : n in N, , x_i in A, , sum_{i=1}^n|lambda_i| leq 1 ight}.

References

*

ee also

*vector (geometric), for vectors in physics
*Vector field


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