We prove that an $\epsilon$-approximate fixpoint of a map $f:[0,1]^d\rightarrow [0,1]^d$ can be found with $\mathcal{O}(d^2(\log\frac{1}{\epsilon} + \log\frac{1}{1-\lambda}))$ 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'24] from the $\ell_\infty$-case to all $\ell_p$-metrics. Previously, all query upper bounds for $p\in [1,\infty) \setminus \{2\}$ were either exponential in $d$, $\log\frac{1}{\epsilon}$, 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 $\textsf{FP}^{dt}$. 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).

翻译：我们证明，若映射 $f:[0,1]^d\rightarrow [0,1]^d$ 关于某个 $p\in [1,\infty)\cup\{\infty\}$ 的 $\ell_p$ 度量是 $\lambda$ 收缩的，则可通过 $\mathcal{O}(d^2(\log\frac{1}{\epsilon} + \log\frac{1}{1-\lambda}))$ 次对 $f$ 的查询找到一个 $\epsilon$ 近似不动点。这将 Chen、Li 和 Yannakakis [STOC'24] 最近关于 $\ell_\infty$ 情形的结果推广至所有 $\ell_p$ 度量。此前，对于 $p\in [1,\infty) \setminus \{2\}$ 的所有查询上界，在 $d$、$\log\frac{1}{\epsilon}$ 或 $\log\frac{1}{1-\lambda}$ 中至少有一个是指数级的。Chen、Li 和 Yannakakis 还展示了在 $\ell_\infty$ 情形下如何确保所有对 $f$ 的查询都位于有限粒度的离散网格上。我们为 $\ell_1$ 情形提供了这样的舍入方法，从而将 $\ell_1$ 情形的一个适当定义版本置于 $\textsf{FP}^{dt}$ 中。为证明我们的结果，我们引入了 $\ell_p$ 半空间的概念，并推广了离散几何中的经典中心点定理：对于任意 $p \in [1, \infty) \cup \{\infty\}$ 及任意质量分布（或点集），我们证明存在一个中心点 $c$，使得由 $c$ 和法向量定义的每个 $\ell_p$ 半空间至少包含 $\frac{1}{d+1}$ 比例的质量（或点）。