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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.09525 |
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| _version_ | 1866912274955894784 |
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| author | Zanotti, Leo |
| author_facet | Zanotti, Leo |
| contents | 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 $p=Ω(n^{d+1-\frac{c}{\sqrt{\log_2(n)}}})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09525 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds on the Number of Pieces in Continuous Piecewise Affine Functions Zanotti, Leo Combinatorics Computational Geometry Discrete Mathematics 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 $p=Ω(n^{d+1-\frac{c}{\sqrt{\log_2(n)}}})$. |
| title | Bounds on the Number of Pieces in Continuous Piecewise Affine Functions |
| topic | Combinatorics Computational Geometry Discrete Mathematics |
| url | https://arxiv.org/abs/2503.09525 |