Simple solid regions in a coordinate direction and their cyclic coordinate projection

Coordinates on ${\mathbb{R}}^3$ are named $x,y,z$ for the indices $0,1,2$ of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. For each $k\in \{x,y,z\}$ the cyclic coordinate projection ${\pi }_k:{\mathbb{R}}^3\to {\mathbb{R}}^2$ drops the $k$th coordinate and keeps the other two in cyclic order:

${\pi }_x(p)=(p_y,p_z),{\pi }_y(p)=(p_z,p_x),{\pi }_z(p)=(p_x,p_y)\mathrm{.}$

A simple description of a solid in the direction $k$ is a quadruple $(k,D,{\gamma }_1,{\gamma }_2)$ in which $D\subseteq {\mathbb{R}}^2$ is compact, Jordan measurable and has nonempty interior, and ${\gamma }_1,{\gamma }_2:D\to \mathbb{R}$ are continuous with ${\gamma }_1\le {\gamma }_2$ on $D$ and ${\gamma }_1<{\gamma }_2$ on the interior of $D$. The simple solid region it describes is

$E=\{p\in {\mathbb{R}}^3:{\pi }_k(p)\in D,{\gamma }_1({\pi }_k(p))\le p_k\le {\gamma }_2({\pi }_k(p))\}\mathrm{.}$

The set $D$ is the base, ${\gamma }_2$ the upper graph function and ${\gamma }_1$ the lower graph function of the description. A solid is simple in the direction $k$ when some such description of it is supplied; the description is part of the data and is not inferred from the set $E$.

Writing ${\sigma }_k$ for the cyclic permutation of A cyclic permutation of the coordinates of ${\mathbb{R}}^3$ preserves Jordan measurability and integrals, so that ${\sigma }_k(p)=({\pi }_k(p),p_k)$, the image ${\sigma }_k[E]$ is exactly the solid between the graphs of ${\gamma }_1$ and ${\gamma }_2$ over the base $D$ in the sense of A solid between continuous graphs over a compact Jordan base. That set is compact and Jordan measurable by A solid between continuous graphs over a compact Jordan base is Jordan measurable and integrates by vertical sections, and ${\sigma }_k^{-1}$ is again a cyclic coordinate permutation, so $E$ is compact and Jordan measurable as well; integration over $E$ is that of The Riemann integral of a bounded function over a bounded Jordan measurable set, and interiors, closures and boundaries are those of Interior, closure, boundary, limit point, isolated point and dense subset of a metric space.

• Weak inequality on the base, strict inside. The graphs are allowed to meet on $\mathrm{\partial }D$, so a vertical section of $E$ over a boundary point of the base may be a single point; that is what lets a ball be described in every direction, since its two hemispherical graph functions agree exactly on the equatorial circle. The strictness on the interior of $D$ is what makes the interior of $E$ nonempty and is used where the outward normal is identified.

• The cyclic order is not cosmetic. With ${\pi }_y(p)=(p_z,p_x)$ rather than $(p_x,p_z)$, each ${\sigma }_k$ has determinant $1$ and each coordinate of an oriented area vector is the Jacobian determinant of the matching projection; taking the surviving coordinates in increasing order would reverse both signs in the case $k=y$ and no statement on this page would hold uniformly in $k$.

• Nonempty interior of the base. A base with empty interior need not make $E$ a graph: a line-segment base with ${\gamma }_1<{\gamma }_2$ produces a vertical rectangle. It does, however, make $E$ three-dimensionally content zero and makes the strictness condition on the interior vacuous. Requiring nonempty interior keeps every simple solid region a genuine solid. The boundary of $D$ has content zero by A bounded set in ${\mathbb{R}}^m$ is Jordan measurable iff its boundary is null, equivalently of content zero, which is what makes the base the closure of its interior up to a negligible set in the arguments that follow.