This paper explores the connection between classical isoperimetric inequalities, their directed analogues, and monotonicity testing. We study the setting of real-valued functions $f : [0,1]^d \to \mathbb{R}$ on the solid unit cube, where the goal is to test with respect to the $L^p$ distance. Our goals are twofold: to further understand the relationship between classical and directed isoperimetry, and to give a monotonicity tester with sublinear query complexity in this setting. Our main results are 1) an $L^2$ monotonicity tester for $M$-Lipschitz functions with query complexity $\widetilde O(\sqrt{d} M^2 / \epsilon^2)$ and, behind this result, 2) the directed Poincar\'e inequality $\mathsf{dist}^{\mathsf{mono}}_2(f)^2 \le C \mathbb{E}[|\nabla^- f|^2]$, where the "directed gradient" operator $\nabla^-$ measures the local violations of monotonicity of $f$. To prove the second result, we introduce a partial differential equation (PDE), the directed heat equation, which takes a one-dimensional function $f$ into a monotone function $f^*$ over time and enjoys many desirable analytic properties. We obtain the directed Poincar\'e inequality by combining convergence aspects of this PDE with the theory of optimal transport. Crucially for our conceptual motivation, this proof is in complete analogy with the mathematical physics perspective on the classical Poincar\'e inequality, namely as characterizing the convergence of the standard heat equation toward equilibrium.

翻译：本文探讨了经典等周不等式、其有向形式与单调性测试之间的内在联系。我们研究定义在单位立方体$[0,1]^d$上的实值函数$f : [0,1]^d \to \mathbb{R}$，其测试目标基于$L^p$距离。本研究具有双重目标：进一步阐释经典等周不等式与有向等周不等式之间的关联，并在此框架下构建具有亚线性查询复杂度的单调性测试算法。主要研究成果包括：1）针对$M$-Lipschitz函数的$L^2$单调性测试器，其查询复杂度为$\widetilde O(\sqrt{d} M^2 / \epsilon^2)$；2）作为该结果的理论基础，我们建立了有向庞加莱不等式$\mathsf{dist}^{\mathsf{mono}}_2(f)^2 \le C \mathbb{E}[|\nabla^- f|^2]$，其中“有向梯度”算子$\nabla^-$用于度量函数$f$的局部单调性违反程度。为证明第二个结果，我们引入了一个偏微分方程——有向热方程，该方程可将一维函数$f$随时间演化为单调函数$f^*$，并具备诸多理想的分析特性。通过将该偏微分方程的收敛特性与最优传输理论相结合，我们最终推导出有向庞加莱不等式。从概念动机的角度看，该证明过程与经典庞加莱不等式在数学物理学中的诠释完全对应——即将其视为标准热方程向平衡态收敛的特征描述。