relative interior

Let S be a subset of the n-dimensional Euclidean space n. The relative interior of S is the interior of S considered as a subset of its affine hull Aff(S), and is denoted by ri(S).

The differencePlanetmathPlanetmath between the interior and the relative interior of S can be illustrated in the following two examples. Consider the closed unit square


in 3. Its interior is , the empty setMathworldPlanetmath. However, its relative interior is


since Aff(I2) is the x-y plane {(x,y,0)x,y}. Next, consider the closed unit cube


in 3. The interior and the relative interior of I3 are the same:



  • As another example, the relative interior of a point is the point, whereas the interior of a point is .

  • It is true that if TS, then int(T)int(S). However, this is not the case for the relative interior operator ri, as shown by the above two examples: I2I3, but ri(I2)ri(I3)=.

  • The companion concept of the relative interior of a set S is the relative boundary of S: it is the boundary of S in Aff(S), denoted by rbd(S). Equivalently, rbd(S)=S¯-ri(S), where S¯ is the closure of S.

  • S is said to be relatively open if S=ri(S).

  • All of the definitions above can be generalized to convex sets in a topological vector spaceMathworldPlanetmath.

Title relative interior
Canonical name RelativeInterior
Date of creation 2013-03-22 16:20:07
Last modified on 2013-03-22 16:20:07
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 52A07
Classification msc 52A15
Classification msc 51N10
Classification msc 52A20
Defines relative boundary
Defines relatively open