Exact maximum number of pieces for a fixed number of affine components

Determine the maximum possible number of pieces of a continuous piecewise affine function on \(\mathbb{R}^d\) having a prescribed number \(n\) of distinct affine components.

Background

For a continuous piecewise affine function, the number of pieces $p$ and the number of distinct affine components $n$ provide two different measures of complexity, with the elementary inequality $p\geq n$. The paper studies the largest possible value of $p$ as a function of $n$ for functions on $\mathbb{R}^d$.

The paper proves an upper bound $p=O(n^{d+1})$ and constructs examples with $p=\Omega\!\left(n^{d+1-c/\sqrt{\log_2 n}}\right)$, showing that the upper bound is nearly sharp but does not determine the exact maximum.