# minimizing the integral of a function over square sets.

Hi!

I'm interested in some problems, but to be honest i'm not sure of the field they belong to.

Let $h(x,y)$ be a bivariate function on $X^2$, where $X$ is some nice topological space (for instance $[0,1]$, or $[0,1]^2$.)

1. What is $$\inf_A \int_{A\times A} h(x,y)dxdy?$$ where $A$ belongs to the class of measurable sets (I think one can use other classes of sets, like the closed sets, or the open sets, etc...).

2. What is $$\inf_Y \sum_{x,y\in Y} h(x,y)?$$ where $Y$ belongs to the class of finite sets. What is $$\inf_Y \sum_{x\neq y\in Y} h(x,y)?$$

I think the answers are difficult, but if one has any idea or an idea of a link with something else, it is already a lot.

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Here is a possible connection: if you take the $Y$ in the discrete case to be subsets of some finite set then you have a clique problem in graph theory, where $h$ represents edge weights. – Niels Diepeveen Oct 6 '11 at 21:47
Yes it is interesting as well, could you tell me more or indicate a reference? – kaleidoscop Oct 7 '11 at 16:07