Relative interior

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In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces.

Formally, the relative interior of a set S (denoted ) is defined as its interior within the affine hull of S.[1] In other words,

where is the affine hull of S, and is a ball of radius centered on . Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior.

For any nonempty convex sets the relative interior can be defined as


See also[edit]


  1. ^ Zălinescu, C. (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing  Co., Inc. pp. 2–3. ISBN 981-238-067-1. MR 1921556. 
  2. ^ Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 47. ISBN 978-0-691-01586-6. 
  3. ^ Dimitri Bertsekas (1999). Nonlinear Programming (2 ed.). Belmont, Massachusetts: Athena Scientific. p. 697. ISBN 978-1-886529-14-4. 

Further reading[edit]