Bounds on the Number of Pieces in Continuous Piecewise Affine Functions

Published 12 Mar 2025 in math.CO, cs.CG, and cs.DM | (2503.09525v2)

Abstract: The complexity of continuous piecewise affine (CPA) functions can be measured by the number of pieces $p$ or the number of distinct affine functions $n$. For CPA functions on $\mathbb{R}^d$, this paper shows an upper bound of $p=O(n^{d+1})$ and constructs a family of functions achieving a lower bound of $p=\Omega(n^{{d+1-\frac{c}{\sqrt{\log_2(n)}}})$.}