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A piecewise linear function agrees with an affine function on each member of a finite or locally finite family of subsets partitioning its domain of a function. On the real line the pieces are intervals; in higher dimensions each piece can be a polyhedron. Here the conventional word “linear” permits a constant offset within each piece.

For example, the absolute value function satisfies $|x|=-x$ for $x\le 0$ and $|x|=x$ for $x\ge 0$. Both branches agree at zero, so the resulting function is continuous, although its derivative changes there. Joining sampled values by linear interpolation gives another common construction. If the partition pieces meet, their formulas must agree on overlaps to define a single function; allowing separate pieces across a jump is a convention that should be stated.

A polyhedron with two congruent parallel polygonal bases whose corresponding vertices are joined by parallel edges. A right prism has edges between the bases perpendicular to their planes. The optical use of a transparent prism is a separate application of this geometric shape.

A pyramid is a polyhedron with a polygon as its base and triangular faces joining the base edges to one apex outside the base plane. If its base area is $A$ and perpendicular height is $h$, its volume is $Ah/3$. At height $z$ above the base, similarity gives cross-sectional area $A(1-z/h{)}^2$. Integrating this area from $0$ to $h$ proves the formula.

Sinai Robins coauthored a text on integer-point enumeration with Matthias Beck. Their exposition develops Ehrhart polynomials, generating functions and Ehrhart reciprocity for polyhedra. The subject gives a precise bridge between discrete counting and continuous geometry: the leading growth of lattice-point counts records volume.^[ref]

For the faces of a closed polyhedron, take outward unit normal vectors. Applying the divergence theorem to an arbitrary constant vector field $c$ givesSince $c$ is arbitrary, the vector areas sum to zero. This is an oriented statement; the positive scalar surface areas do not cancel. The same proof works for every piecewise smooth closed boundary to which the divergence theorem applies.

A corner of a polyhedron, stored with its spatial coordinates and incidences with boundary edges and faces. In a closed manifold boundary representation, the incident faces form a single cyclic fan. The vertices of a polyhedron and edges define a graph, but geometric embedding and valid face incidence require additional information.