Naimark's problem

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 $\left|{\lambda }_1\right|+\left|{\lambda }_2\right|\le 1$ 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 then for any subset A of a vector space one can define its absolutely convex hull to 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

$\text{absconv}A=\{\sum _{i=1}^n{\lambda }_i{x}_i:n\in \mathbb{\mathbb{N}},x_i\in A,\sum _{i=1}^n|{\lambda }_i|\le 1\}$.

References

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See also

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