Example 2.2 from Boyd and Vandenberghe's Convex Optimization:

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The textbook says that the interior of the unit square

$$\{x \in \mathbb{R}^3 | -1 \leq x_1 \leq +1, -1 \leq x_2 \leq +1, x_3 = 0\}$$ is empty.

The interior of a set is defined as the union of all open subsets of the set. It seems to me that in this case, the interior should be the same as the relative interior noted in the example (the unit square, excluding the boundary). Can someone explain why this is not the case?

  • $\begingroup$ because they are just lines ;) try to draw a open ball inside a line ;) $\endgroup$ Jul 12 '19 at 20:38
  • $\begingroup$ It is a bit misleading to call this set unit square. Usually, the unit square is taken to be $(0,1)^2$ (closed or not) in $\mathbb R^2$. $\endgroup$
    – daw
    Jul 15 '19 at 7:59

Because every open ball centered at a point of the unit square contains points such that their third coordinate is not $0$. Of course, no such point belongs to the unit square.

  • $\begingroup$ If there were no mention of "$x_3$" then you would be working in two dimensions and the interior would be non-empty. Because of the "$x_3= 0$ you are working in three dimensions. As others have said a ball of radius r centered at any point in the square will contain points where $x_3$ is not 0. $\endgroup$
    – user247327
    Jul 12 '19 at 20:58

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