We study the connection between directed isoperimetric inequalities and monotonicity testing. In recent years, this connection has unlocked breakthroughs for testing monotonicity of functions defined on discrete domains. Inspired the rich history of isoperimetric inequalities in continuous settings, we propose that studying the relationship between directed isoperimetry and monotonicity in such settings is essential for understanding the full scope of this connection. Hence, we ask whether directed isoperimetric inequalities hold for functions $f : [0,1]^n \to \mathbb{R}$, and whether this question has implications for monotonicity testing. We answer both questions affirmatively. For Lipschitz functions $f : [0,1]^n \to \mathbb{R}$, we show the inequality $d^{\mathsf{mono}}_1(f) \lesssim \mathbb{E}\left[\|\nabla^- f\|_1\right]$, which upper bounds the $L^1$ distance to monotonicity of $f$ by a measure of its "directed gradient". A key ingredient in our proof is the monotone rearrangement of $f$, which generalizes the classical "sorting operator" to continuous settings. We use this inequality to give an $L^1$ monotonicity tester for Lipschitz functions $f : [0,1]^n \to \mathbb{R}$, and this framework also implies similar results for testing real-valued functions on the hypergrid.

翻译：我们研究了有向等周不等式与单调性测试之间的联系。近年来，这一联系为离散域上函数单调性测试的研究带来了突破性进展。受连续情况下等周不等式丰富历史的启发，我们提出，在连续情境下研究有向等周性与单调性之间的关系，对于全面理解这一联系至关重要。因此，我们探究了有向等周不等式是否对函数$f : [0,1]^n \to \mathbb{R}$成立，以及这一问题是否对单调性测试具有意义。我们对这两个问题均给出了肯定回答。对于Lipschitz函数$f : [0,1]^n \to \mathbb{R}$，我们证明了不等式$d^{\mathsf{mono}}_1(f) \lesssim \mathbb{E}\left[\|\nabla^- f\|_1\right]$，该式通过$f$的“有向梯度”度量来界定其与单调函数的$L^1$距离。我们证明中的一个关键要素是$f$的单调重排，它将经典的“排序算子”推广到了连续情境。利用这一不等式，我们为Lipschitz函数$f : [0,1]^n \to \mathbb{R}$设计了一个$L^1$单调性测试器，且该框架还蕴含了在超网格上测试实值函数的类似结果。