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Consider a smooth function $$ f: [0,\pi] \to \mathbb{R}.$$

Let $X,Y$ be independent, identically distributed random variables that live on $S^1$. We assume that they are absolutely continuos.

We can then compute the following quantity

$$I(p(x)) = \mathbb{E}\left[f\left(d(X,Y)\right)\right]$$

where $d$ is the distance on the circle and $p$ is the density of $X$.

My question now is for which choice of distribution does this quantity achieve it's extrema?

That is, what is for example

$$ \arg \max I(p(x))?$$

I found these related questions, but they all seem to be considered with functions $f$ which only take positive values.

Apex angle of a triangle as a random variable

Expected absolute difference between two iid variables

I am not sure if for example the proof strategy employed by Sangchul Lee in the answers to these questions can be carried over to the more general case, since he uses a rather particular condition on $\int f((d(x,y)) q(x) q(y) dx dy$.

a_student
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    It is interesting to know how such an extremal problem arose. – Botnakov N. Jan 26 '23 at 16:04
  • This seems like it would be heavily dependent on $f$... not sure what a general solution would look like – John Don Jan 26 '23 at 16:35
  • @JohnDon Sure, but I was hoping that there might maybe be some nice characterization dependent on the Fourier transform of $f$ – a_student Jan 27 '23 at 10:13
  • @BotnakovN. It is essentially the same as looking at a problem of the sort $\int \int p(x) k(x,y) p(y) dx dx$ where $k(x,y) = f(x-y)$ is a kernel implementing convolution with $f$. So if it were not for the restriction to probability measures, quite a standard question I would say. Then the restriction to $S^1$ seemed easier since there are already some questions out there. – a_student Jan 27 '23 at 10:16

1 Answers1

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As you mentioned in your comment this is a question about minimising the energy $$ E(\rho):=\int_{\mathbb{S}^1} f* \rho\, \, \mathrm{d}\rho \, , $$ over all probability measures on the circle. The above energy is referred to as the interaction energy and there is a zoo of literature on its properties. See, for example, the following papers which study the problem on $\mathbb{R}^d$: https://arxiv.org/pdf/2202.09237.pdf, https://arxiv.org/abs/2107.05079, https://arxiv.org/abs/1607.08660.

If $f$ is smooth and even, minimisers concentrate of global minima of $f$, i.e. they are atomic measures with mass on the global minima of $f$. The Euler--Lagrange condition for critical points of this energy is $\nabla f * \rho= \mathrm{const.}$ on each connected component of the support of $\rho$.

If $f$ has a singularity things get more complicated and you can end up with minimisers whose support has fractional dimension (as illustrated in the papers above).

almosteverywhere
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  • Thank you for the literature, this problem seems harder than I thought it would be. Regarding your answer: What does it mean for the minimisers to concentrate on global minima of $f$? If the minimum of $f$ is at $z \neq 0$, then one needs to put mass at values $x,y$ such that $x-y =z$? But this then also implies that there will be mass at other values, in particular at $0$, so if $f$ is to large at 0, shouldn't this counteract that solution? – a_student Feb 03 '23 at 16:14