# Properties of special rectangle (measure)

Let $I$ be a special rectangle in $\mathbb{R}^n$, and denote $\lambda(A)$ the measure of $A$. Prove that the following conditions are equivalent:

a) $\lambda(I)=0$

b) $I^{\circ}=\emptyset$ (i.e., the interior of $I$ is empty)

c) $I$ is contained in an affine subspace of $\mathbb{R}^n$ having dimension smaller than $n$. (An affine subspace is any set of the form $\{x_0+x|x\in E\}$, where $x_0\in\mathbb{R}^n$ is fixed and $E$ is a subspace of the vector space $\mathbb{R}^n$. Its dimension is equal to the dimension of $E$.)

The implication $a\implies b$ isn't too bad (using the definition if $I=[a_1,b_1]\times...\times[a_n,b_n]$ then $\lambda(I)=(b_1-a_2)...(b_n-a_n)$. So we conclude $a_i=b_i$ for some $i$. And since $I^{\circ}$ is open, there can't be anything contained in it). I also see how c) makes sense (at least intuitively), but not sure how to show it formally. Thank you.

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$a)\implies b)$ Already proved.
$b)\implies c)$ Since $I^\circ=\varnothing$, then $a_{i_0}=b_{i_0}$ for some $i_0\in\{1,\ldots,n\}$. Denote $c=a_{i_0}=b_{i_0}$ then, in particular, $x_{i_0}=c$ for all $x\in I$. Consider affine subspace $X=c e_{i_0}+\mathbb{R}^{n-1}=$$\{x\in\mathbb{R}^n:x_{i_0}=c\}$$\subset \mathbb{R}^n$. Therefore $I\subset X$.
$c)\implies a)$ Proof ad absurdum. Assume that $\lambda(I)>0$, though $I\subset X$ for some affine subspace $X$ of dimension less than $n$. Since $\lambda(I)>0$ then $b_i>a_i$ for all $i\in\{1,\ldots,n\}$. Let $r=\min_{i\in\overline{1,n}}(b_i-a_i)/2$ and $a$ be the center of $I$, then $\operatorname{Ball}(a,r)\subset I$. Since $X$ is an affine subspace, then $X=x_0+E$ with $\dim E<n$. Consider any $z\in E^\perp$ (note $E^\perp\neq \{0\}$, because $\dim E< n$). Without loss of generality we may assume that $\Vert z\Vert=r/2$. As $a\in I$, then $a+z\in \operatorname{Ball}(a,r)\subset I\subset X$. Since $a\in I\subset X$ and $a+z\in X$, then $z=(a+z)-a\in X-X\subset E$. Contradiction, because $z$ is non zero vector in $E^\perp$.
Hi user 70520, in my opinion you waste your money because the asked result is a very particular case of the following result: let $K$ be a convex compact subset of $\mathbb{R}^n$. Then a),b) and c) are equivalent properties of $K$. –  loup blanc Sep 21 '13 at 21:44