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 $\int_{x\in\mathcal{X}} |f(x)|^2\,\mathrm{d}x$?
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1$\begingroup$ Take $f=C$, $C$ constant. Then the integrals can be made arbitrarily large by choosing $C$ appropriately. $\endgroup$– Liviu NicolaescuCommented Jul 27, 2018 at 11:02
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$\begingroup$ Agree. But what if we add additional constraints like $ f(0)=0$? $\endgroup$– JeffCommented Jul 28, 2018 at 0:14
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$\begingroup$ I have added this assumption. $\endgroup$– JeffCommented Jul 28, 2018 at 23:30
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1$\begingroup$ How is this either algebraic geometry or differential geometry? $\endgroup$– Nik WeaverCommented Jul 29, 2018 at 2:10
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1$\begingroup$ @NikWeaver Because it uses $^2$ and $dx$ LOL. $\endgroup$– Fan ZhengCommented Jul 30, 2018 at 1:08
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1 Answer
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If $f:[0,1]^n\to\mathbb{R}$ is $L$-Lipschitz and $f(0)=0$, then $|f(x)|\le L\|x\|$, which obviously implies that the function $x\mapsto L\|x\|$ is the maximizer for both problems.
answered Jul 29, 2018 at 0:10
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1$\begingroup$ This example generalizes to $f(x)=|x|$ on $[0,1]^n$. $\endgroup$ Commented Jul 29, 2018 at 0:13
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$\begingroup$ @Fan Zheng it was a typo. fixed $\endgroup$ Commented Jul 29, 2018 at 11:32