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、其有向类比以及单调性检验之间的联系。我们研究定义在实心单位立方体$[0,1]^d$上的实值函数$f : [0,1]^d \to \mathbb{R}$，目标是在$L^p$距离下进行检验。我们的目标有两个：进一步理解经典等周不等式与有向等周不等式之间的关系，以及在此设定下给出具有次线性查询复杂度的单调性检验器。我们的主要结果包括：1) 针对$M$-Lipschitz函数的$L^2$单调性检验器，其查询复杂度为$\widetilde O(\sqrt{d} M^2 / \epsilon^2)$；2) 在此结果背后，我们得到了有向Poincaré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$\nabla^-$度量了函数$f$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（PDE）——有向热方程，该方程将一维函数$f$随时间演化为单调函数$f^*$，并具有许多理想的分析性质。通过将该PDE的收敛性质与最优输运理论相结合，我们得到了有向Poincaré不等式。对我们的概念动机至关重要的一点是，这一证明与经典Poincaré不等式的数学物理视角完全类似，即将其刻画为标准热方程趋向平衡态的收敛过程。