Let $f:{\bf R}^n\to {\bf R}$ ($n\geq 2$) be a $C^1$ function. Is it true that $$\sup_{x\in {\bf R}^n}f(x)=\sup_{x\in {\bf R}^n}f(x+\nabla f(x))\hskip 3pt ?$$
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1$\begingroup$ Notice that this cannot be reduced to the almost trivial 1-dimensional case by looking at the line through $x$ and $x+\nabla f(x)$. $\endgroup$– Yaakov BaruchNov 29, 2019 at 8:18
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5$\begingroup$ Note that $f\left(x+\nabla f(x)\right) \leq \sup_{x\in \mathbb{R}^n}f(x)$ and hence $\mathrm{sup}_{x\in \mathbb{R}^n}f\left(x+\nabla f(x)\right) \leq \mathrm{sup}_{x\in \mathbb{R}^n}f(x)$. Now assume that there exists $x_0$ such that $f(x_0) = \mathrm{sup}_{x\in \mathbb{R}^n}f(x)$, then $\nabla f(x_0) = 0$ and hence $f(x_0+\nabla f(x_0)) = f(x_0) = \mathrm{sup}_{x\in \mathbb{R}^n}f(x)$. So you are interested in the case when $f$ does not attain its global maximum. Am I right or I am lost? $\endgroup$– SlupNov 29, 2019 at 15:58
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1$\begingroup$ @Slup: This is an interesting observation. You prove that the statement holds if the sup is a max, but as you pointed out yourself, there are other scenarios. For example, your argument also works for $f(x-\nabla f)$, but in this case, the claim is false ($f=|x|^2/2$). $\endgroup$– Christian RemlingNov 29, 2019 at 16:36
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5$\begingroup$ A trivial remark: no such function can be bounded from below; moreover, $\liminf (f(x) / |x|^2) \leqslant -\tfrac{1}{2}$. Indeed: otherwise, for every $p \in \mathbb{R}^n$, $f(x) + (x - p)^2/2$ would have a global minimum at some $q \in \mathbb{R}^n$, and so $0 = \nabla f(q) + (q - p)$, that is, $p = q + \nabla f(q)$. In other words, the range of $x + \nabla f(x)$ is all of $\mathbb{R}^n$. $\endgroup$– Mateusz KwaśnickiDec 2, 2019 at 7:26
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1$\begingroup$ In any case my suggestion does not need to be bounded below. It can be replaced with $f(x,y)=\sin\big(\sqrt{x^2+y^2}-2\arctan(\frac{y}{x})\big)(e^{x^2+y^2}-1)$. tinyurl.com/qq3o8kb . I really don't expect this to work, but the (naive) idea is that any point on the crest or close to it be thrown off the tangent of the crest and down the very steep slope. (And one has to check where all the other points go too, but at least the region where $f$ is below some large enough finite bound is connected, as it would need to be.) $\endgroup$– Yaakov BaruchDec 3, 2019 at 10:44
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