Relative interior

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. Intuitively, the relative interior of a set contains all points which are not on the "edge" of the set, relative to the smallest subspace in which this set lies.

Formally, the relative interior of a set "S" (denoted $ext\{relint\}(S)$) is defined as its interior within the affine hull of "S". In other words,:$ext\{relint\}(S)\; =\; \{\; x\; in\; S\; :\; exists\; epsilon\; >\; 0,\; (N\_epsilon(x)\; cap\; ext\{aff\}(S))\; subset\; S\; \},$where $ext\{aff\}(S)$ is the affine hull of "S", and $N\_epsilon(x)$ is a ball of radius $epsilon$ centered on $x$. Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior.

References

* cite book
last = Boyd
first = Stephen
coauthors = Lieven Vandenberghe
title = Convex Optimization
publisher = Cambridge University Press
date = 2004
location = Cambridge
pages = p. 23
url = http://www.stanford.edu/boyd/cvxbook.html
id = ISBN 0 521 83378 7