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Let $h$ be a nonnegative, smooth and convex function on $[0,1]$ and let $f(x,y):[0,1]\times[0,1]\rightarrow[0,1]$ with $f(x,y)=f(y,x)$ and $f$ continuous. Suppose I fix $r>0$ and demand that

$$ |||f|||:=\int_0^1\int_0^1\int_0^1f(x,y)f(y,z)f(z,x)dxdydz=r. $$

What can be said about

$$ \inf_{|||f|||=r} \int_0^1\int_0^1 h(f(x,y))dxdy,$$

where the infimum is taken over all such $f$ of the above characterization? In particular, is there a way of obtaining the minimizer $f$ in an analytic way? If not, I would love to be pointed toward specific literature on such extremizations.

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