Query-Efficient Fixpoints of ℓp-Contractions

We prove that an $\varepsilon$-approximate fixpoint of a map $f:[0,1]^{d} \rightarrow[0,1]^{d}$ can be found with $\mathcal{O}\left(d^{2}\left(\log \frac{1}{\varepsilon}+\log \frac{1}{1-\lambda}\right)\right)$ queries to f if f is $\lambda$-contracting with respect to an $\ell_{p}$-metric for some $p \in[1, \infty) \cup\{\infty\}$. This generalizes a recent result of Chen, Li, and Yannakakis [STOC 2024] from the $\ell_{\infty}$-case to all $\ell_{p}$ metrics. Previously, all query upper bounds for $p \in[1, \infty) \backslash\{2\}$ were either exponential in $d, \log \frac{1}{\varepsilon}$, or $\log \frac{1}{1-\lambda}$. Chen, Li, and Yannakakis also show how to ensure that all queries to f lie on a discrete grid of limited granularity in the $\ell_{\infty}$-case. We provide such a rounding for the $\ell_{1}$-case, placing an appropriately defined version of the $\ell_{1}$-case in FPdt. To prove our results, we introduce the notion of $\ell_{p}$-halfspaces and generalize the classical centerpoint theorem from discrete geometry: for any $p \in[1, \infty) \cup\{\infty\}$ and any mass distribution (or point set), we prove that there exists a centerpoint c such that every $\ell_{p}$-halfspace defined by c and a normal vector contains at least a $\frac{1}{d+1}$-fraction of the mass (or points).