Exercise 1.1.9 (Compact convex polytopes are Jordan measurable)

First note the $d=1$ case is trivial since convex sets are intervals and compactness guarantees the polytope is a closed and bounded interval, which is an elementary box in ${ℝ}^1$. Thus, assume $d>1$. Since the polytope is bounded, let $B$ be an elementary box containing the polytope. Furthermore, since the polytope is the intersection of finitely many half-spaces, let $H$ be one such half-space. It suffices to show the set $X=B\cap H$ is Jordan measurable, since intersecting these sets together for each $H$ will retrieve the polytope, and their intersection will be Jordan measurable by Exercise 1.1.6(1).

Let $H=\left\{x\in ℝ\mid a\cdot x\le c\right\}$ for some $c\in ℝ$ and $a\in {ℝ}^d$, not all coordinates equal to $0$. Without loss of generality, assume $a_1>0$ (the case $a_1<0$ is similar). Additionally, by Exercise 1.1.6(6) (translation invariance), we may assume $B$ has its first interval of the form $(0,M]$ for some $M>0$. We identify ${ℝ}^{d-1}$ with the subset $\{0\}\times {ℝ}^{d-1}$ of ${ℝ}^d$.

Now define the function $P:{ℝ}^{d-1}\to ℝ$ by

where $x_0=(x_2,x_3,\dots ,x_d)$. Clearly, the graph of $P$ is the hyperplane boundary of $H$. Now let $B_0={ℝ}^{d-1}\cap B$ and define the function $f:B_0\to ℝ$ by

Since constants and $P$ are continuous, we know $f$ is continuous. Define the set

where $y=(y_1,y_2,\dots ,y_d)$ and $y_0=(y_2,\dots ,y_d)$. We claim that $X=Y$.

If $x\in X$, then $x\in B$ and $x\in H$. Since $x\in H$, we know

which implies $x_1\le P(x_0)$. Since $x_1\in (0,M]$, we know $0<x_1\le \min \{M,P(x_0)\}$, which implies $0<x_1\le f(x_0)$. Thus, $x\in Y$, so $X\subseteq Y$.

Conversely, if $y\in Y$, then $y_0\in B_0$ and $0<y_1\le f(y_0).$ Since $f(y_0)>0$, we must have

Thus, $0<y_1\le M$ and $y_1\le P(y_0)$, the latter implying $a_1{y}_1+\dots +a_d{y}_d\le c$. Thus, $y\in H$. Since $y_1\in (0,M]$ and $y_0\in B_0$, we know $y\in B$. Thus, $y\in X$, so $Y\subseteq X$.

Therefore, $X=Y$. Since $Y$ is the region under the graph of the function $f$ minus the graph of the $0$ function, we know by Exercise 1.1.7 that $X$ is Jordan measurable.

Pedagogical notes: The intuition for the function $f$ can be attained from looking at the $d=2$ case and attempting to define in all cases the continuous function forming the top of the region $X$. For the $a_1<0$ case, one should choose the first interval of $B$ to be of the form $[0,M)$ and let $Y$ be the space between $f$ and the constant function $M$.