Linear algebra, affine space, and floor function

Question

My question is mostly: is there a name for this kind of things. I am mostly interested by finding book or articles about what follows, but without even a word or a name, it is quite hard to search for information.

Let $F$ be the smallest class of functions $f(x_1,\dots,x_n)$ which contains the functions: -real constants -projection on one component (that is, $x_i$ for some $1\le i\le n$, -$\lfloor f\rfloor$, for some $f\in F$, -$f_1+f_2$, for $f_1,f_2\in F$ -$cf$ for some $c\in\mathbb R$ and $f\in F$

As stated by Dmitri Zaitsev, the functions in $F$ are piecewise affine functions. But this description is too general, there are piecewise affine function which lacks the periodicity provided by the floor function. Therefore, I would like a name for $F$, or at least, to know which kind of property is satisfied by functions of $F$.

(Of course, those functions are interpreted on $\mathbb R$, but if it helps it could be $\mathbb Q$, as I don't know any other field where $\lfloor\rfloor$ is defined, apart from $\mathbb Z$ where the question becomes trivial.)

Answer 1

Such functions are special cases of the so-called piecewise affine (or not very accurately "piecewise linear" as their pieces are affine, not linear, see here for an explanation of the difference between affine and linear).

Formally, a function is piecewise affine if its domain of definition can be decomposed into a disjoint union of polytopes, on each of which it is affine. Here the polytopes are sets defined by finitely many affine equations and inequalities.

To get a better understanding, it is instructive to look at the simplest case of a floor $g(x) = \lfloor f(x)\rfloor$ of a single affine function $f$. Then $g$ is constant on the affine stripes $n\le f(x) < n+1$, and the level set $g=const$ defines one of the stripes or empty set, depending on the constant.

Slightly more generally, $g(x) = f_0(x) + a\lfloor f(x)\rfloor$ is affine on each stripe, whose values on each two stripes differ by a constant. Now the level sets (where the constructed function is constant) are parallel affine hyperplanes inside the stripes (for functions in general position).

More generally, for an arbitrary function $f(x)$ as in the question, the level sets of each $\lfloor f_j(x)\rfloor$ is the family of parallel stripes $n\le f_j(x) < n+1$. Then $f$ is affine on each polytope obtained by intersecting the stripes. Again, values of $f(x)$ on each two such intersections must differ by a constant. Hence, functions constructed in this way are very special among piecewise affine ones.

Given the above interpretation, each level set $g(x)=const$ consists of a family of parallel hyperplanes on each polytope.

More abstractly, the floor function $\lfloor t \rfloor$ can be replaced by any piecewise constant function valued in any set, leading to the same conclusion.