Absolutely convex set

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


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 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[edit]

The light gray area is the Absolutely convex hull of the cross.

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

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

See also[edit]


  • Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53. Cambridge University Press. pp. 4–6.