# Boyd & Vandenberghe, example 2.2 — interior and relative interior points

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 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?

• 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 $S$ May 12, 2018 at 13:38
• @percusse I don't understand; are you addressing something in my explanation? May 12, 2018 at 13:49
• 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 $x$. May 12, 2018 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 $S$. May 12, 2018 at 17:31
• Added to The List Apr 7, 2021 at 14:04

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 $R^3$".