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The curve $\Gamma$ in $\mathbb{R}^2$ is defined by a continuous and monotonically increasing function $f(x)\in\text{C}[0,1]$, where $f(0)=0$, $f(1)=1$.

Let $(X,Y)$ is jointly and uniformly distributed on the curve $\Gamma$. Here we consider the expectation values of $X$ and $Y$, i.e., $E(X)=\int_{\Gamma} \frac{x}{M} ds$ and $E(Y)=\int_{\Gamma}\frac{y}{M}ds$, where $M$ is the length of the curve $\Gamma$.

I want to know the necessary and sufficient conditions of $E(X)$ and $E(Y)$ when such a curve exists. Dual problems are whether such the curve exists with given values of $E(X)$ and $E(Y)$ and how to construct this curve. For example, based on my own obeservation, the curve does not exist when both $E(X)$ and $E(Y)$ is stricly less than 0.5.

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  • $\begingroup$ So, you want to prescribe the barycenter of the curve, right? How is this related to calculus of variations or convex optimization? Nothing is being optimized whatsoever... Why is it that such a cruve cannot exist if $E(X)$ and $E(Y)$ are less than $\frac 12$? $\endgroup$
    – leo monsaingeon
    May 19, 2020 at 8:15

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