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We study the connection between directed isoperimetric inequalities andmonotonicity testing. In recent years, this connection has unlockedbreakthroughs for testing monotonicity of functions defined on discretedomains. Inspired the rich history of isoperimetric inequalities in continuoussettings, we propose that studying the relationship between directedisoperimetry and monotonicity in such settings is essential for understandingthe 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 formonotonicity testing. We answer both questions affirmatively. For Lipschitzfunctions $f : [0,1]^n \to \mathbb{R}$, 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 ofits "directed gradient". A key ingredient in our proof is the monotonerearrangement of $f$, which generalizes the classical "sorting operator" tocontinuous settings. We use this inequality to give an $L^1$ monotonicitytester for Lipschitz functions $f : [0,1]^n \to \mathbb{R}$, and this frameworkalso implies similar results for testing real-valued functions on thehypergrid.

Directed Poincar\'e Inequalities and $L^1$ Monotonicity Testing of Lipschitz Functions