Absolutely convex set

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A set C in a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (circled), in which case it is called a disk.


A set is absolutely convex if and only if for any points in and any numbers satisfying the sum belongs to .

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


See also[edit]


  • Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 4–6. 
  • Narici, Lawrence; Beckenstein, Edward (July 26, 2010). Topological Vector Spaces, Second Edition. Pure and Applied Mathematics (Second ed.). Chapman and Hall/CRC.