A piecewise linear function can be described in different forms: as an arbitrarily nested expression of $\min$- and $\max$-functions, as a difference of two convex piecewise linear functions, or as a linear combination of maxima of affine-linear functions. In this paper, we provide two main results: first, we show that for every piecewise linear function there exists a linear combination of $\max$-functions with at most $n+1$ arguments, and give an algorithm for its computation. Moreover, these arguments are contained in the finite set of affine-linear functions that coincide with the given function in some open set. Second, we prove that the piecewise linear function $\max(0, x_{1}, \ldots, x_{n})$ cannot be represented as a linear combination of maxima of less than $n+1$ affine-linear arguments. This was conjectured by Wang and Sun in 2005 in a paper on representations of piecewise linear functions as linear combination of maxima.

本文提供了两个主要结果：首先，我们证明了对于每个分段线性函数，都存在具有最多 $n+1$ 个参数的 $\max$-函数的线性组合，我们给出了计算它的算法；其次，我们证明了分段线性函数 $\max(0, x_{1}, \ldots, x_{n})$ 不能用小于 $n+1$ 个仿射线性参数的最大值的线性组合来表示。

用小数组表示分段线性函数