## Relative Interior

Oct 14 2016

This is a series of notes on convex optimization, covering technical details/backgrounds as I study it for the first time. The classic text Convex Optimization, will be heavily referenced.

The mathematical definition $$\textbf{relint }C = \{x \in C | B(x, r) ∩ \textbf{aff } C ⊆ C \text{ for some }r > 0\}$$, stated in English, is the set of points that are the centers of norm balls with some positive radius such that these balls' intersections with $$\textbf{aff }C$$ are always contained in $$C$$. We then define the relative boundary of a set $$C$$ as $$\textbf{cl } C \setminus \textbf{relint } C$$, where $$\textbf{cl } C$$ is the closure of $$C$$. The notion of closure is in many ways dual to the notion of interior--the closure of $$C$$ is the smallest closed set containing $$C$$, whereas the interior of $$C$$ is the biggest open set within $$C$$. The boundary of $$C$$ is the set of points in $$\textbf{cl }C$$ not belonging to the interior of $$C$$.

A point is in the $$\textit{interior}$$ of a set if you can draw a small open ball around it which is itself contained in the set. The concept of the ''interior'' of a set can also be defined relative to the ''ambient space'' around it (more precisely, its affine hull)--that's the idea behind the relative interior. This is most useful when the interior of a set is empty, but its relative interior captures the points we want to work with. For example, the unit circle in $$\mathbb{R}^2$$, $$A=\{x ∈ \mathbb{R}^2 | x_1^2 + x_2^2 = 1\}$$, has the unit open disk $$B=\{x ∈ \mathbb{R}^2 | x_1^2 + x_2^2 < 1\}$$ as its interior; but the unit circle sitting on x-y plane in $$\mathbb{R}^3$$, i.e.,$$E=\{x ∈ \mathbb{R}^3 | x_1^2 + x_2^2 = 1, x_3=0\}$$, has empty interior, by definition. The open disk $$D=\{x ∈ \mathbb{R}^3 | x_1^2 + x_2^2 < 1, x_3=0\}$$ is not the interior of $$E$$, since any norm ball with center located in $$D$$ will necessarily capture some points of $$\mathbb{R}^3$$ above and below $$D$$ that are not in $$E$$. But $$D$$ is kind of an interior (and a far more interesting one than the empty interior)--we declare it the relative interior of $$E$$, relative to $$\textbf{aff }E = \{x ∈ \mathbb{R}^3|x_3=0\}$$; then we're back to the $$\mathbb{R}^2$$ case of $$B$$ being the interior of $$A$$.

Also see book example 2.2, p.23 of a relative boundary.

Other resources: