Jordan Regions III
1. Properties of Jordan Regions
Theorem
If \(E_1,\ldots,E_N\) are Jordan regions, then \(E_1 \cup \cdots \cup E_N\) is a Jordan region and
\[{}\text{Vol}(E_1 \cup \cdots \cup E_N){}\]
\[{}\leq \text{Vol}(E_1) + \cdots + \text{Vol}(E_N). {}\]
Proof
Theorem
If \(E_1, E_2\) are Jordan regions, then \(E_1 \cap E_2\) is a Jordan region and \(E_1 \setminus E_2\) is a Jordan region. Moreover,
\[ \text{Vol}(E_1) = \text{Vol}(E_1 \cap E_2) + \text{Vol}(E_1 \setminus E_2). \]
Proof
2. Example: Bounded Convex Sets are Jordan Regions
Theorem
Every bounded convex \(E\) set in \({\mathbb R}^n\) is a Jordan region.
Proof
Lemma
Let \(e_1,\ldots,e_n\) be the standard unit coordinate vectors. If \(K\) is a convex set in \({\mathbb R}^n\) which contains vectors of the form \(\pm c^{\pm}_j e_j + \delta^{\pm}_j\) for each \(j\) and each choice of \(\pm\), where \(c^{\pm}_j \geq n\) and each \(\delta^{\pm}_j\) is a vector all of whose coordinates have magnitude at most \(1/2\), then the set \(K\) contains \([-1/2,1/2]^n\).
Proof
It suffices to show that all vertices of \([-1/2,1/2]^n\) are contained in \(K\). The proof is by induction on dimension. The case \(n=1\) is trivial: any point in \([-3/2,-1/2]\) and any point in \([1/2,3/2]\) have convex hull containing all of \([-1/2,1/2]\). So suppose \(n \geq 2\). Consider the value of \(\theta\) for which
\[ v:= (1-\theta) (\pm c_j^{\pm} e_j + \delta_j^{\pm}) + \theta ( c^+_n e_n + \delta_n^+) \]
has \(+1/2\) in its \(n\)-th coordinate position, where \(j < n\). This \(\theta\) satisfies
\[ \theta = \frac{\frac{1}{2} - (\delta_j^{\pm})_n}{c^+_n + (\delta_n^+)_n - (\delta_j^{\pm})_n}\]
where \((\delta_j^{\pm})_n\) is the \(n\)-th coordinate of \(\delta_j^{\pm}\) and so on. Because \(c_n^+ \geq 2\) and \(|(\delta_n^+)_n|, |(\delta_j^{\pm})_n| \leq 1/2\), the numerator and denominator can never be negative and the denominator in particular is at least \(1\). As a function of \((\delta_j^{\pm})_n\), \(\theta\) is decreasing when \((\delta_j^{\pm})_n \leq c^+_n + (\delta_n^+)_n - (1/2)\), which includes the entire interval \([-1/2,1/2]\). Thus
\[ 0 \leq \theta \leq \frac{\frac{1}{2} + \frac{1}{2}}{c^+_n + (\delta_n^+)_n + \frac{1}{2}}. \]
Now as a function of \((\delta_n^+)_n\), the numerator is minimized when it equals \(-1/2\), so
\[ 0 \leq \theta \leq \frac{1}{c_n^+} \leq \frac{1}{n}. \]
So the vector \(v\) must equal
\[ \pm \tilde c_j^{\pm} e_j + \tilde \delta_j^{\pm}\]
for \(\tilde c_j^{\pm} \geq (n-1) c_j^{\pm}/n \geq (n-1)\) and \(\tilde \theta_j^{\pm}\) again having coordinate entries bounded by \(1/2\) in magnitude (because it's a convex combination of two such vectors). This means that \([-1/2,1/2]^{n-1} \times \{1/2\}\) is contained in \(K\), and by symmetry and convexity, all of \([-1/2,1/2]^n\) belongs to \(K\) as well.