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I am reading Boyd & Vandenberghe's Convex Optimization. As I have not yet studied topology, I am having some trouble fully understanding example 2.2 on page 23:

Example 2.2 $ \ $ Consider a square in the $(x_1,x_2)$-plane in $\mathbf{R}^3$, defined as $$C=\{x\in\mathbf{R}^3\mid-1\leq x_1\leq1,\,-1\leq x_2\leq1,\,x_3=0\}.$$ Its affine hull is the $(x_1,x_2)$-plane, i.e., $\mathbf{af}\!\mathbf{f}\,C=\{x\in\mathbf{R}^3\mid x_3=0\}$. The interior of $C$ is empty, but the relative interior is $$\mathbf{relint}\,C=\{x\in\mathbf{R}^3\mid-1<x_1<1,\,-1<x_2<1,\,x_3=0\}.$$ Its boundary (in $\mathbf{R}^3$) is itself; its relative boundary is the wire-frame outline, $$\{x\in\mathbf{R}^3\mid\max\{|x_1|,|x_2|\}=1,\,x_3=0\}.$$

Specifically, I was not fully understanding the concepts of interior and relative interior points. So I began conducting research, which resulted in me posting an answer. To ensure that I understood the concepts, especially within the context of example 2.2, I decided to "explain" the idea to myself, which resulted in the following explanation:

We have a square in the $(x_1, x_2)$-plane in $\mathbb{R}^3$, and, since $x_3 = 0$, the affine hull of $C$, $\mathbf{af}\!\mathbf{f}\, C$, is all of the $(x_1, x_2)$-plane, $\mathbf{af}\!\mathbf{f}\, C = \{ x \in \mathbb{R}^3 | x_3 = 0 \}$, as was stated.

The issue is that we had $B_{\delta}(x) \not \subseteq S$ (where $B_{\delta}(x)$ is the ball of radius $\delta$ entered at the point $x$), for any $\delta > 0$, where the subset of points $S$ is considered as a subset of $\mathbb{R}^3$. This can be thought of as the interior relative to $\mathbb{R}^3$. Instead, we now consider the subset of points $S$ as a subset of $\mathbf{af}\!\mathbf{f}\,C = \{ x \in \mathbb{R}^3 | x_3 = 0 \}$. This can be thought of as the interior relative to the affine hull.

Note that, when $S$ was considered as a subset of $\mathbb{R}^3$, we would have the ball $B_{\delta}(x) \subseteq S$ for any $\delta > 0$, where $S \subseteq \mathbb{R}^3$ is a set of points. Assuming our distance function $d$ for the associated metric space $(\mathbb{R}^3, d)$ is just the Euclidean distance, the ball $B_{\delta}(x)$ would literally just be a ball. However, when $S$ is considered as a subset of the affine hull $\mathbf{af}\!\mathbf{f}\,C$, we would have the ball $B_{\delta}(x) \subseteq S$ for any $\delta > 0$, where $S \subseteq \mathbf{af}\!\mathbf{f}\,C = \{ x \in \mathbb{R}^3 | x_3 = 0 \}$ is a set of points. Assuming our distance function $d$ for the associated metric space $(\mathbf{af}\!\mathbf{f}\,C, d)$ is just the Euclidean distance, the ball $B_{\delta}(x)$ would be the open disc. This is why the relative interior of $C$, $\mathbf{relint}\,C$, is $\underline{not}$ empty, unlike the interior of $C$ -- because the ball $B_{\delta}(x)$ is now the open disc.

I would greatly appreciate it if people could please take the time to review my understanding by ensuring that everything I wrote above is correct. Are there any errors? Can I improve my explanation?

Rodrigo de Azevedo
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The Pointer
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  • Think of the neighborhoods; there are no 3D open sets around any point in $C$ because $x_3$. But in terms of relative interior there is in 2D. Because that is allowed by the definition of $S$ – percusse May 12 '18 at 13:38
  • @percusse I don't understand; are you addressing something in my explanation? – The Pointer May 12 '18 at 13:49
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    Tangential comment, but a simple definition of the relative interior of a set $S \subset \mathbb R^n$ is just ${ x \in S \mid B_\delta(x) \cap \text{aff}(S) \subset S \text{ for some } \delta > 0}$. Here $\text{aff}(S)$ is the affine hull of $S$ and $B_\delta(x)$ is the open ball of radius $\delta$ centered at $x$. – littleO May 12 '18 at 17:25
  • @littleO Yes, good addition. This was stated in the book, too. In other words, the relative interior is the set of all points $x$ in the set of points $S \subset \mathbb{R}^n$, such that $x$ is an element of the ball of radius $\delta$, centered at the point $x$, AND the affine hull of $S$. – The Pointer May 12 '18 at 17:31
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    Added to The List – Rodrigo de Azevedo Apr 07 '21 at 14:04

1 Answers1

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It is all clear if you remember that in topology interior is always relative. We have a topological space $X$ and a set $A\subset X$, and then we define the interior of $A$ using open balls in $X$. So in fact to be precise it should be called interior of $A$ in $X$. The same applies to boundary.

Now the "relative interior" and "interior" in the sense of Boyd are simply "interior in $R^2$" and "interior in $R^3$".

Michal Adamaszek
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  • Thanks for the response, Michal. What you say is clear to me, and it sounds very similar to what I wrote in my explanation. So would you say that what I wrote as my explanation is correct? Based on your response, It seems like I'm understanding this properly. – The Pointer May 12 '18 at 17:24
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    @ThePointer yes I think this is the same as your explanation but I tried to put it in a more general context. – Michal Adamaszek May 12 '18 at 18:38