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 $lambda\_1\; x\_1\; +\; lambda\_2\; x\_2$belongs 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

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ee also

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