Example in textbook

How is the interior of set C empty in this example? There is definitely more than one $x \in C$ such that $B(x,\epsilon) \subset C$.


According to the definition in Boyd/Vandenberghe's book, $x \in C$ is an interior point of a set $C \subseteq \mathbb{R}^n$ if

$$ \{ y \in \mathbb{R}^n: \| y-x \|_2 \leq \epsilon\} \subseteq C $$

for some $\epsilon > 0$. There is no such $x$ in your example, since any neighborhood around any point in $C$ (in $n$-dimensional space) contains at least one point outside of $C$.

  • $\begingroup$ I totally forgot the n-dimensional aspect of the ball. Thank you for reminding. $\endgroup$ – SPRajagopal Sep 22 '14 at 6:24

We recall interior and relative interior of a set $C\subset\mathbb{R}^n$.

  • The interior of $C$ is $$ \textbf{int }C=\{x\in C: \exists\varepsilon>0 \text{ such that } \; B(x;\varepsilon)\subset C\}, $$ where $B(x;\varepsilon):=\{y\in\mathbb{R}^n: \|x-y\|_2<r\}$;

  • The relative interior of $C$ is $$ \textbf{relint }C=\{x\in C: \exists\varepsilon>0 \text{ such that } \; B(x;\varepsilon)\bigcap \textbf{aff }C\subset C\}, $$ where $\textbf{aff }C$ is the smallest affine set containing $C$.

Since $C=\{x\in\mathbb{R}^3:-1\leq x_1\leq 1, -1\leq x_2\leq 1, x_3=0\}$, there is no ball in $\mathbb{R}^3$ contained in $C$ and $$ \textbf{aff } C=\{x\in\mathbb{R}^3:x_3=0\}. $$ It follows that $$ \textbf{int }C=\emptyset, \quad \textbf{relint }C=\{x\in\mathbb{R}^2:-1<x_1<1, -1<x_2<1, x_3=0\} $$ and the relative boundary of $C$ is $$ \overline{C}\setminus\textbf{relint }C=C\setminus\textbf{relint }C=\{x\in\mathbb{R}^3:\max\{|x_1|, |x_2|\}=1, x_3=0\}. $$


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