relative interior
Let be a subset of the -dimensional Euclidean space . The relative interior of is the interior of considered as a subset of its affine hull , and is denoted by .
The difference between the interior and the relative interior of can be illustrated in the following two examples. Consider the closed unit square
in . Its interior is , the empty set. However, its relative interior is
since is the - plane . Next, consider the closed unit cube
in . The interior and the relative interior of are the same:
Remarks.
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As another example, the relative interior of a point is the point, whereas the interior of a point is .
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It is true that if , then . However, this is not the case for the relative interior operator , as shown by the above two examples: , but .
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is said to be relatively open if .
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All of the definitions above can be generalized to convex sets in a topological vector space.
Title | relative interior |
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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 |