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Let $x \in \mathcal{X} = [0,1]^n$, and $f(x)$ be an $L$-Lipschitz function. Let $f(0)=0$. What is (the exact or a non-trivial upper bound on) $\int_{x\in\mathcal{X}} |f(x)|\,\mathrm{d}x$?What about $\int_{x\in\mathcal{X}} |f(x)|^2\,\mathrm{d}x$?
Let $x \in \mathcal{X} = [0,1]^n$, and $f(x)$ be an $L$-Lipschitz function. What is (the exact or a non-trivial upper bound on) $\int_{x\in\mathcal{X}} |f(x)|\,\mathrm{d}x$?What about $\int_{x\in\mathcal{X}} |f(x)|^2\,\mathrm{d}x$?
Let $x \in \mathcal{X} = [0,1]^n$, and $f(x)$ be an $L$-Lipschitz function. Let $f(0)=0$. What is (the exact or a non-trivial upper bound on) $\int_{x\in\mathcal{X}} |f(x)|\,\mathrm{d}x$?What about $\int_{x\in\mathcal{X}} |f(x)|^2\,\mathrm{d}x$?
A non-trivial upper bound on the integral of Lipschitz functions over a bounded support
Let $x \in \mathcal{X} = [0,1]^n$, and $f(x)$ be an $L$-Lipschitz function. What is (the exact or a non-trivial upper bound on) $\int_{x\in\mathcal{X}} |f(x)|\,\mathrm{d}x$?What about $\int_{x\in\mathcal{X}} |f(x)|^2\,\mathrm{d}x$?
